The answer is cleared up if you pull back and look at the phenomenon of friction for what it is - it is an interaction between two material interfaces which dissipates kinetic energy. In the case of the dissipative interface between solid materials, it seems to go like the following: One material is sitting there with its surface molecules or atoms in a largely minimum energy state. Along comes another material which has its surface molecules (or atoms) very close to the original material's surface (they are touching). The close proximity of the two molecules or atoms induces a rearrangement of the charge structure around each of the surface (interface) molecules or atoms which tend to make them stick together by electric forces. As the materials slide past each other and are no longer in 'contact', the surface molecules/atoms will decay back into their initial energy states (at Maxwell-Boltzmann distribution). This action results in thermal excitement (heating) in those surface molecules which will conduct into the material and generally radiate away. (Aside: You can see something similar to this by the way magnets fall slowly when their moving magnetic fields are allowed to induce eddy currents in a metal pipe they are falling through, even when falling in vacuum and not touching the walls of the tube. The magnetic field is dissipating its energy in the generation of these eddy current loops as it moves past. Note that the current loops go away after the magnet has passed by. I hope that you have seen this demo or one similar to it.)

I would need to consult with a chemist on this, but I suspect that this is basically what an adhesive is. One material is particularly susceptible to charge rearrangement - put it in paste form so that it has a large surface area (to contact as much other atoms as possible) and there you go, glue. Again, I am not sure about this one so check it out for yourself. For that matter, science is based on "check it out for yourself," so if you really want to be sure...

Anyway, if the interface materials do not move relative to each other, they will still have the interface molecules/atoms change their electronic structure, but they will simply stay that way and thus no energy dissipation - you just need some energy to get the surfaces together to begin with (gravitational potential energy will do, they will come into contact and deform slightly to accomplish the interface contact fusing).
But there is more! There is also the loss of energy in all solid materials due to deformation. This will lead to atoms/molecules that are not in direct contact to also have to rearrange their electronic charge distributions not to mention the direct energy loss in just pushing around a mass or material in order to get it in to some new shape.

There is also energy loss in the form of acoustic (sound) waves in the materials and maybe extra losses when small chunks of material are broken off completely and send flying. I really couldn't say which of these possible losses of energy is the main one as it clearly depends on the situation. I would not expect similar car performance between driving on diamond and driving on silly putty. Finally, most often solid-surface friction is simply due to material roughness physically pushing on material attached to the other surface. The above arguments are useful only for explaining friction between very smooth surfaces. If the surfaces are rough, you can easily imagine that they will mesh in some places, fitting together like toothed gears (just with randomly sized and spaces teeth!).

However, for viscous (drag) friction there is a different, simpler mechanism at work to dissipate the energy. Here the gas is too rarified to cause significant energy loss due to the electronic charge redistribution described above. You are simply using up energy pushing the mass of fluid material in front of the motion forward and out of the way. There may well be much more to viscous fiction than this, but it serves for our purposes.

So lets take this way of looking at it and see if we can answer some of the problems you were having: It now seems clear that the assertion that 'friction opposes the direction of motion' is close but too simplistic (as you pointed out). A better statement would be 'friction opposes the direction of actual or potential sliding motion along the contacting surfaces'. Lets look at you individual questions and see how this revised statement works. "For example, when a car is moving at constant velocity, the normal force is counteracted by the force of gravity, and so no net force in y-direction, hence, no acceleration in the vertical direction. The drag force (of say, the wind) acts to oppose the motion of the vehicle. Thus, if the car is moving at constant velocity, doesn't the friction between the tires and the road act in the direction of motion?" Yes.

It seems that you and your students have stumbled onto a mostly unexplained (at least to anyone's satisfaction) phenomenon in physics. As far as your examples, I would say the statement that friction opposes motion only applies to kinetic friction, not static friction. Both of your examples demonstrate static friction, though there is motion. Consider the case of the tire: at the point of contact, the tire is not moving relative to the road. The reason wheels work is that the static friction sticks the tire to the road. The tire is exerting a force backward and the friction is directed forward in opposition so, though there is no motion, the friction is opposed to the force. In the case of the object in the truck bed, it is not moving relative to the surface of the truck, meaning the friction is static also. If it were moving relative to the truck you could make the same argument, of course, because then you would have kinetic friction. However, the friction force opposes the motion of the object relative to the contact surface in this case also. I also want to point out that the equation f = u N is *NOT* always right. It is an empirical formula developed that seems to describe what is going on in a crude way. It can easily be broken. One example of a way that it can be broken is to attach an object to a force spring scale and try to pull the object with a constant velocity. The equation tells you that the force should be proportional to the normal force but you should see very quickly that, rather than move smoothly, the object jerks forward in spurts. You can see this on the force meter by watching the force fluctuate. I don't know how easy it would be to see this, but the friction force should also depend on the speed of pulling, which is not captured in the equation either.

An interesting thing to measure then is how the force fluctuations depend on the surface area in contact between the object and surface. Another thing you could look at is seeing how the fluctuations are affected by dragging the object up an inclined plane. I have no idea what you will discover! The way I understand friction is that there are numerous microscopic stick-slip events. In other words, a small portion of the objects surface will stick for a poorly understood reason.

Some elastic energy will build as you keep pulling, then suddenly that region will pop forward to the next sticking point. Somehow, these microscopic events average out to what you observe when you pull an object across a surface. There is a Scientific American article which I have not read but might be useful: "Friction at the Atomic Scale" by Jacqueline Krim in the October 1996 issue. The article is about nanotribology, defined as the study of friction at the atomic level.

The statement: "Friction always opposes the direction of motion" applies to SLIDING friction.

In order for the car to roll forward, there must be ROLLING friction between the tires and the road. This is more like STATIC friction, in that for an "ideal" tire the point of contact with the road has ZERO instantaneous velocity relative to the road. So that point at the bottom in contact is not sliding. In idealized problems involving rolling things, we call this "rolling without slipping". So, you are correct: in this case friction acts to pull the tires forward. Try putting a bicycle wheel on a rod and spinning it. Lower it slowly so that it just barely touches the ground - you will feel the sliding friction oppose the rolling of the tire. Put it firmly on the ground and you will feel it roll forward. You can feel the point when it changes from sliding friction to rolling friction.

Another example, a stationary box is on the bed of a truck accelerating from rest, and is at rest in the frame of reference of the moving truck. Isn't the box accelerating, in the reference frame of the ground, and so, the net force must be from the friction between the surface of the bed of the truck and the box. Again, this is in the forward direction, but yes, this time, the statement 'Yes - friction IS pulling the box along with the truck, but this is clearly STATIC friction that is holding the box on the truck. In order for static friction to keep the box at rest in the frame of the truck, it must be sufficient to hold the box in place.

You can try this by putting a string around a book and putting a block or smaller book on top. Pull the book slowly and gently along the table top, and static friction between the books will keep the smaller one on top. Accelerate the bottom book with a quick jerk of the string, and the smaller book will slip backward, relative to the acceleration. In this latter situation, the relative motion of the upper book with respect to the lower book is opposed by sliding friction. This will simulate the situation you pose about the box on the truck bed.

I have to ask: What text book are you using????
Try UNIVERSITY PHYSICS by Young and Freedman. There is a good explanation about static vs. kinetic friction in chapter 5. This book is approved for AP Physics by the college board. Any good text should go into detail about the difference between kinetic and static friction, and have a graph, and a picture of the contact surfaces, showing the little asperities in the surfaces where the bonding takes place. Friction is, on the microscopic level, actually an electromagnetic interaction. Of the four "fundamental forces" of nature (strong and weak nuclear, electromagnetic, and gravitational) friction is actually due to electromagnetic interaction between the atoms at the contact surfaces. Oh, and what is the nature of the relationship between the normal force and friction. Is the equation f = u N always true? Where does this equation come from? How do scientists actually perform experiments to measure the coefficient of friction values?

f = u N for kinetic (sliding) friction, but f <= u N for static friction. Pull a book slowly with a string. At first the book won't move - static friction prevents two objects from sliding; but when the force of the pull exceeds the limit of the "bonds" between the microscopic asperities in the the two surfaces to hold together, then the little teensy weensy bonds break, and the book suddenly jerks forward, and if you keep pulling, it will slide along the table, with kinetic friction opposing the motion. You can simulate this by clasping your hands together, and pulling them apart with the strength of your arms. At a certain point, your fingers can't keep your hands together and your hands will have to release. That is what is going on at the contact surfaces. For the same reason we get earthquakes - the mantle underneath is in constant motion, dragging the crust along with it. The crustal plates stick together at the boundaries and faults, but when the stress is too great, the friction between the two sides of a fault is not sufficient to keep it together, and the fault ruptures due to the pulls from below.

Here is a classic experiment to do with students - have them put a string around a book and pull with a hand-held scale - they can READ the scale in Newtons as their applied force increases (hence the opposing force of static friction also increases) until suddenly the scale drops, and then returns to a constant level, but lower than the maximum before sliding began.

The normal force is the contact force between two surfaces, and is always perpendicular to the contact surfaces.

Well -- friction is actually a collective name for several differing phenomena ranging from fluid drag to sound and shear wave generation. The low down is that empirical force equations like F =u N are approximate and apply only in special circumstances. For example, as you mention, static friction coefficients are quite different from moving (dynamic coefficients). This is the reason that so many friction phenomena create periodic sounds as the point in contact interactively impacts the surface, deforms, and then breaks free. Many processes involving friction deform the surfaces (either permanently or not) -- thus introducing elasticity with possible non-normal force responses.

Friction with a lubricant often has very non-linear empirical equations --

eg

F = u f(N) + kV + gV² where V is the normal velocity and f is some derived function.

Friction usually does act to oppose the direction of motion -- in your car example, to oppose the drag of air, force must be exerted in the direction of the cars motion -- however, the friction of the road is acting to oppose the slipping of the tires in the direction opposite to the drag. This does not contradict the statement since it is the tires motion at the point of contact that is being opposed, and this motion is in the opposite direction from the cars motion. (Consider a large enough drag that the tires overcome static friction and break free -- now they are clearly moving in the direction that is being opposed by friction.

As to your second question about "normal" force, this is clearly an approximation only good for fairly rigid bodies in contact, but even there it is dicey. There are almost always other forces involved -- potentially in directions different from either the normal force or the direction of motion, due to the small deformation of the surfaces.

In an old style LP record, the needle rode a wave of semi-fluid vinyl which eventually restored to the original shape quite well. However, the needle was accelerated laterally by microscopic level deviations to produce a reasonable sound reproduction.