This is the kind of question for which you really should consult your physical therapist, rather than a web-site. If you have healed well from the surgery and are in good shape, then yes, you can ride a mountain bike with a knee replacement. But I do not know what kind of condition you are in.
A road bike is faster on the road than a mountain bike. That's all
Mountain bike shoes are not required to be able to ride a mountain bike. You can wear any good athletic shoe. Wearing an actual mountain bike shoe is helpful however to keep your feet from slipping from the peddles on rugged terrian but it not required in any way.
It is a sport were you ride down a mountain and take jumps on a bike.
You ride up and off a ramp
because the traction on the mountain bike tires have more of a grip to be able to ride on a mountain when a road isn´t that hard to ride on and to need more traction.
Online, mountain bike parts can be bought from many online suppliers. Jenson USA and Go-ride are two examples of sites that dedicate their inventory to mountain bike parts.
As a matter of fact that's just what I have and I am a keen bike rider so one has to take it easy.
Mountain bikes have a taller scale than 20 inch bikes, but are better suited for outdoor ruggedness.
I would suggest a bike with a small frame and a long seat post if you want a low seat for jumping, but a larger frame if you ride mountain and street more.
A bike is a bike is a bike, can you ride one you can ride them all - after a fashion. MTBs in particular are rather easy to handle as the riding position is fairly upright. In terms of difficulties of merely riding down the road I'd say that a road bike for the same given size is more difficult to ride as it will have the rider more stretched out and hunched over, but it's not like that is hugely challenging either. The hardest bikes to ride are probably the recumbents, particularly the ones with waist steering. Those tends to take some getting used to.
No, it is illegal to ride a dirt bike on the street but you can ride it on a trail or dirt bike track.
There is a group called the Northeast Tennessee Mountain Bike Association. They are a group of mountain bikers located in Tennessee that ride together. More information can be found at www.ntmba.org/.