This is an interesting question and really deals with the most fundamental laws of the universe. A concrete example of this idea is if you have a ball that is sitting on a table and you push it off. Once the ball if no longer on the table, it can lower its energy by falling to the ground which will always happen. Though, it’s important to remember that it’s not the earth that favors things with the lowest energy, but the entire universe. Arguably the two most important laws that govern the universe are the first and second laws of thermodynamics. The first law says energy is conserved, which is important to know, but doesn’t really answer your question. The answer to your question is a result of the second law which says that theentropy of the universe is always increasing. Entropy is a way to characterize every system in the universe and can roughly be thought of a measure of how much you know about a system before measuring it explicitly. The simplest example is if you have a bag of marbles and every marble is red, the system has an entropy value of zero. If half of the marbles are red and other half are blue, you have the maximum entropy value because if you pick a marble at random, you are the least sure of which color marble you are going to pick. There are a number of arguments that can be used to suggest that the entropy of the universe is always increasing, but they can be complicated. Everyday observations like how if you add food dye and water, they will mix on their own are a consequence of the second law.

So what does this half to do with energy? Well as it turns out, energy and entropy are intimately related. In most any system, if you add energy, the entropy will increase. Using calculus you can show that if entropy is at a maximum as is required by the second law, then energy is at a minimum. Therefore, energy is minimized in a system, though it is constant in the entire universe. In a sense, it’s really the mathematics, rather than intuition that drive this conclusion. Though one way to think about it is in the situation with the ball on a table, what would happen if the ball didn’t fall down and just stayed there or floated up? It would probably be hard for a consistent universe to develop without energy minimization.

That is a good question. The short answer is that things with the lowest energy are the most stable. For matter (molecules, minerals, ions, etc), it is easiest to explain using an analogy:
If you place a ball on a hill it will roll down the hill to the bottom. If you place a marble on the side of a bowl, it will also roll to the bottom. This is because the bottom of the hill, and the bottom of the bowl are more stable positions for the ball; the potential energy from gravity is lower at the bottom. Similarly if you try to balance a rock on a stick, it will fall over unless you get it just right, because it is a less stable position than just lying on the ground.

For chemistry, the principle is essentially the same. Chemical reactions happen if the products are more stable than the reactants, which means that they have a lower overall energy. I know that energy is an abstract concept, but I hope that this helped!

This is an interesting question, but let's rephrase it a little bit: "Why do systems minimize their energy at equilibrium?" The answer to this question comes back to the second law of thermodynamics, which can be stated in the following way: "The entropy of an isolated system does not decrease" -- that is, it can only stay the same or increase. It turns out that a system at equilibrium will maximize its entropy, which corresponds to a minimization of energy. Let's think a little more deeply about what this means and where it comes from.

We can start by thinking about entropy. Entropy is a measure of the multiplicity of a system, or the number of "configurations" or "microstates" a system can access. This is a rather abstract sentence, so let's use a concrete example to think about it. Consider a glass of water at room temperature that is at equilibrium, with a fixed energy, volume, and number of particles. At any given time, we can take a picture of the water molecules and would find that the molecules take on various positions in the system and orientations (e.g. which way are the H's "pointing"). These different configurations in which we may find the water molecules are the "microstates" available to the system; there is a distribution of microstates (each microstate is accessible with a certain probability), that is consistent with the macrostate (what we see with our eyes). Mathematically, we can write the entropy as:

Now, it turns out that maximizing the entropy corresponds to minimizing the energy. This can be derived from the first law of thermodynamics, by starting from the fundamental relationship
:dE = -TdS - PdV
(a restatement of the first law) and taking a second derivative of the whole equation with respect to some internal degree of freedom. Why a second derivative? Recall from your calculus classes that extremum principles come from second derivatives (if something is concave down, it has a local maximum and its second derivative is less than 0 --
f"(c) < 0,
vice versa for a local minima). What one finds when they go through this process is that the second derivative of S is less than 0 (S" < 0), corresponding to a local maximum, and the second derivative of E is greater than 0 (E" > 0), corresponding to a local minimum.

This is a very complicated question. It is really a question of statistical mechanics, which is a field of physics and math. I'll try to explain as best I can.

Nature actually has no preference or favorites, and this results in systems tending toward the lowest energy. This sounds like it doesn't make any sense but I can help explain this with a simple experiment. First note that energy isn't ever created or destroyed, it is conserved, so it has to go somewhere. In reality, it's often released as heat. So, let's do an experiment to see how nature shares energy and has no preferences, but how this results in preferring the low energy state. You'll need a pencil and paper, a coin to flip, and dice to roll.

In general there is this idea of each bit of matter having different "energy states" where it holds a certain amount of energy. Let's examine this a bit farther. Say for example we have two bits of matter and six bits of energy. (Draw this part out to follow along.) These two bits of matter share the energy. How do these bits of energy get divided up though? Completely randomly. The universe has no preference for where the energy goes. So, for each bit of energy, toss a coin. For heads, place the bit of energy in matter #1. For tails, place it in #2. Do this for all six bits, and look at what you get. It may not be perfectly equal sharing of 3 and 3 bits of energy. Do this several more times and keep track of the results. The most common result will be an even split of 3 and 3 just by chance. This is how nature distributes energy.

Now, think of those same bits of matter. What would happen if we brought those two into contact with four more bits of matter and again let the energy distribute itself randomly? Do this with a dice. Make six bits of energy, and roll the dice to determine where each bit ends up for the 6 bits of matter. If you do this several times, you'll see that the most common result is that each bit of matter has one bit of energy. This is a low energy state compared to the possibility of having 6 energy on a single bit of matter. So even though there's no preference for having the lowest energy, by randomly rolling the dice this is the most common outcome! That's how the world works and why things tend toward lowest energy. All random chance drives it, but because of the way statistics work out, it looks like everything is "trying" to get to the lowest energy.

This is incredibly advanced physics, and I highly encourage you to do the tests yourself to see how this works, and how energy ends up evenly distributing itself. It makes it a lot clearer and really helped me. This marching toward even energy distribution happens trillions of trillions of trillions of times every second. It is actually theoretically what keeps time moving forward instead of backward, since once you have the six molecules with the evenly distributed six energy bits, it won't naturally go back to all the energy being in only two molecules. Pretty cool.

The Second Law of Thermodynamics states that entropy (physical disorder) must increase over time. The most disorderly form of energy is heat, and energy cannot be created or destroyed (that's the First Law of Thermodynamics), so in order to produce heat, the only way to do that is to transform energy from other forms into heat. Lowering the energy state of other things allows the energy in them to be released to become heat.