Often when students (such as myself) learn quantum mechanics for the first time they imagine that particles are little points that zip around, spending most of their time around the peaks of the wavefunction. Let's call this the "whizzing particle" model. This picture is tempting because it explains some experimental facts, for example it explains why we always measure a discrete number of photons and never just a piece of a photon. However, it can't explain many other things we see in experiments, so we have to give it up and find another way to understand things. It also leads us to ask questions just like the one you asked (where does the particle start?) and as far as we know there's no way, even in principle, to answer these questions.

One of the main problems you run into with a whizzing particle model is trying to explain the double slit experiment. If you shine laser light at a small slit in a screen you will see a bright spot behind the screen (more or less). Now imagine cutting a second slit in the screen what happens is that instead of seeing two spots, or one big spot, you see one spot with light and dark fringes. You can see a picture of the results of the experiment here

These patterns (called diffraction patterns) are just what we expect for any wave. The interesting thing is that even if you turn down the intensity on the laser to the point were only a single photon comes out of the laser at a time, you still get this different pattern. Each photon hits a seemingly random point on your detector, but when all of the photons are added up you strangely get the same pattern of fringes. The results looks something like this

Quantum mechanics predicts that particles other than the photon should have similar behavior. This was a radical prediction, but it was confirmed experimentally for electrons in 1927, just a few years after these results were predicted.

Just to be thorough, there are two other important reasons the "whizzing particle" model can't be right. One is the phenomenon of quantum entanglement (entanglement). This is the thing Einstein famously called "spooky action at a distance." The other is called the Aharanov-Bohm effect, and has to do with how charged particles are effected by electric and magnetic fields

So, these experiments motivate us to conclude that the photon (or any particle for that matter) does not exist at a certain point in the wave function. Any attempt that I know of to reduce the particle to something "smaller" than the wavefunction either fails to explain the above experiments or ends up sounding a little bit silly. For example, a theory known as Bohmian mechanics (De-Boglie) says that the particle does exist at a particular point at a particular time, but that no one can ever measure it. Most physicists find it more natural to do away with the notion of exact position rather than to posit an exact position which can never be predicted. Besides it feels a little like bending over backwards to construct a theory in which the particle has an exact location but that is still consistent with the double slit experiment.

So, in a word the answer to both your questions is no. It seems that particles really don't have exact positions and velocities.

Well, a photon really isn't a wave; it's a particle. Photons emitted by sources are waves, but those waves describe the probability distribution of the locations of the photon particles. If you interact with the electromagnetic field generated by the photon, then you've interacted with the particle (thus changing it).

By and large, in quantum mechanics, the wave- like properties of energy and matter are more important if there either is a lot of matter or energy available (many photons, as opposed to just a few), or in terms of the probability distributions of said particles' momentum, position, time, and energy. A lone photon thus behaves more like a particle the location and momentum of which are imprecisely defined, rather than a wave. To get light to behave in a wave-like manner, you need a lot of photons, not just one or a few.

It's somewhat difficult to simultaneously think of photons as a particle and as a wave. The Heisenberg uncertainty principle asserts that you cannot simultaneously know both the precise momentum (~energy) and position. When we treat photons as waves, this formalism is generally concerned with the energy of the wave, where the photon is delocalized and its position is uncertain. Similarly, because the photon is the wave pattern, the position of some ideal localized photon is unknown in this description.