Is it really physically impossible for them to trot and gallop?
No, although some won't and they're usually trained not to. It depends on the breed and the horse. It's the same with canter. Saddlebreds, Icelandics and MFT's all canter; other breeds might not.
What is the difference between the different types of "walk"?
Well, there are actually five different easy gaits. However they're called different things by different breed associations which makes it confusing. Using the most common names, the five gaits are: the foxtrot, running walk, saddle rack, rack, and step pace. The difference between the saddle rack and rack is very small, with the rack being faster. The running walk is literally a fast walk, but technically all the other gaits are different in some way. The step pace is not considered desirable for most breeds, but any breed might do it.
What does the paso feel like to ride?
Which paso? There are four - the paso fino, paso corto, paso largo and paso llano. The fino and corto the same as the saddle rack, and the largo is a true rack. They are all gaits of the Paso Fino. The paso llano is technically the same as the running walk except with an outward swinging of the front legs. It is the main gait of the Peruvian.
I have only ridden the saddle rack/rack but I've heard that each gait feels different. In the rack you feel an up-and-down motion but without a bounce if that make sense. You really feel it in the hips.