First off, let me assure you that I have not yet taken a physics class. Yes, I'm fairly young, so don't assume I'm an expert.

Big Definition: The instantaneous (or nearly so) process of a wizard compressing their physical self (based on canon description of feel) into a one-dimensional speck of energy/magic, and moving at presumably the speed of light to a new destination, and reversing the compression. The 'pop' (or 'crack') is created by the sudden displacement of air. Also, a wizard may 'bring along' an undefined number of others, although it is fairly certain physical contact is needed.

Simple Definition: Teleportation

In the Big Definition, I took liberties in assuming the wizard is magically compressed and moves at the speed of light. It could be wrong, but unless there is a better theory, it is the one I will go with.

My Question:

A)Can the wizard change the position that they arrive in to a different one than they left in?

Ex. Standing up, wizard apparates into a chair, ends up sitting down.

B)Does the wizard take along everything immediately touching him/herself? If so, what about shoes (not touching feet because of socks)?

C)Is the wizard's momentum/inertia/weird science term/current movement affect them after they disapparate, or vice versa?

Ex. Wizard falling from the sky, traveling at ~ 70 km/h, can he use apparation to appear safely on the ground? Or if he tried, would his velocity be transported too, and he would end up as a splat?
Opposite Ex. If Death Eaters tried to apparate onto the Hogwarts Express, would their magic automatically adjust for the differences in speed, or would good ole' Bella be smashed against the wall.

 

I've always imagined it as moving outside the third dimension. If you've read A Wrinkle in Time, it's like that.

The answers to all the questions are really just up to the author, though. You do it however you want to.

Edit: This is probably in the wrong section, though.

 

Slight error with your theory:

If you compress a three-dimensional object with the intent of altering the dimensions, it is more likely to collapse in on itself, thus gain a dimension to become four dimensional. Not reduce and become one dimensional as you suggested -- to do that you'd be expanding, not compressing.

+ As far as inertia goes... I can't be certain. Perhaps you would like to compress your body and find out for yourself.
As for air-friction -- assuming we're moving up the dimension scale here -- there'd be no air resistance. But I'd think that the magic would take care of that factor.

 

Considering appartition from North to South and vice-versa present a change in the speed of the Earth's surface, I would assume that the speed is adapted to the target.

As for what is transported, I'd say that it depends on the wizard's thoughts. If he thinks of his whole self being transported as is, i.e. with clothing, his clothes (and shoes) will be transported along. If he is imagining himself naked, I'd assume that his clothes may be left behind.
(Always under the notion that magic needs a certain amount of visualisation to be used.)

 

These kinds of these can go with many possibilities and theories. Though if i were to believe in something i would have to say that magic is lawless and acts outside of the laws of physics, which possibly has it's own set of rules. Which also might mean the limits of wizards and witches and whatnot. Otherwise there would be millions of powerful wizards running around.

 

Well, I think that what goes with them has to do with their level of concentration. I believe it would be more difficult to leave one's clothing behind than to take it with you. I mean the instructor tells them to picture themselves at their destination. do me a favor, picture yourself right now. in that mental picture, were you dressed? I'm betting you were.

Also, something I think is excellent physics is the sound of apparition. you are suddenly removing a mass from it's volume, leaving behind a vacuum which must imediately be filled with air. moving air causes sound, and such a sudden rush of air would cause a sudden and harsh sound.

as for how it works, I would imagine the "Compressed" feeling could be from a shift into fewer dimensions, but not of themselvs, but of the world around them. Imagine shifting the entire world into one dimension; up-down. there would be no back-forward or left-right. thus when the universe is re-expanded, they simply anchor themselves to their new location.

as for retaining momentum (which technically speaking is how difficult it is to stop a moving object), I don't think one can actually apparate successfully when moving. I mean, think about it. All of the concentration they are told they needed and you expect them to be able to do it in freefall or while running. that's asking to be splinched. although there is the question of "What if they apparate from inside a moving vehicle" again, this is all about how the wizard pictures themself I think. so if they don't picture themselves moving, then I don't think they would be moving. and as for apparating into a chair, I think that's a bit too complicated for most wizards. maybe Dumbles and Voldy can do it.

 

Since Wizards apparently move at instantaneously at the onset of their thought to disapparate, it would be impossible to change position.

I don't mind the vacuum sound theory, but I like to believe that there's a bit of sound from the actual magic as well.

I think you retain momentum. First, momentum is simply the force mass carries in a certain vector. As "force" cannot be physically separated from mass, it would be impossible to simply transport mass without energy.

As for disapparating "out" of something, such as clothes, I suppose it's possible. Just as one can "leave" a body part behind, it's theoretically possible to leave certain objects behind.

Number of people apparated with is probably based on magical ability, as it would likely take more magic to transport more people.

One thing I've always included into my visualizations of apparition is a small, faint circle of smoke that expands and fades away quickly after apparition. It's hard for me to imagine people just dissapearing and appearing without any sort of sign besides pops and cracks.

 

It's hard for me to imagine people disappearing in the first place.