Teleportation doesn't use any sort of in-between space. It's instantaneous movement.
The living body has to be fully teleported before it can teleport again. It can be done against the will of the teleported.
Teleportation Jamming is possible, but simply keeping an area out of view and changing the interior will prevent teleportation. Teleportation only works through air/space not currently occupied by mass. If there is substantial mass in the way it will prevent it from occurring, but if that mass is unstable, it can be displaced and pushed away. Those who can teleport will not usually do so into places they are not familiar with because they risk broken bones or worse. Those who can fly and teleport will tend to be bolder.
It's possible to use teleportation to instantaneously swap places with someone by the method of pushing mentioned above. They can't occupy the same space, but some can swap. This would be employed by body guards most likely.
It works through a process of folding space. Imagine you're a dot in the middle of a piece of flat paper. Teleporting would be somehow folding a corner onto the center of the paper where you are, transferring to the corner where it touches, then letting the paper go back to its normal flat state again.
As for some of the other questions, how the eyes see, how the brain works, and just about anything else like that were once mysteries and relegated to: It just does. All of the mentioned items are considered to be extensions of the body. They do it by will and reflex just as easily as a person opens and closes their hands.
In No Angel, Dallas teleports all over the virtual firing range in order to flank other people and get a shot off before they could see him. The limitations of such a tactic are losing track of where everyone is after the original assessment of locations, and after a few minutes he'd also start suffering vertigo.