Well the good news is that you have not likely transmitted this FIP virus to your cat. I will try to explain, but understand that there is a lot that is unknown about FIP still. The first thing to understand is that it is a mutated version of a common virus, the feline corona virus. Something happens and the virus mutates. Researchers are not convinced that it can be transmitted at that point at all from cat to cat. They suspect that in catteries with a lot of cats, the same corona virus is transmitted, prior to mutating, and then mutates inside the cat, causing FIP. In reasearch settings, they have not been able to transmit FIP from infected cat to uninfected one. I still wouldn't advise trying this with your cat, but with the current knowledge, I can tell you that your cat is likely safe.
Even though they did all of the tests, why were they unable to determine for sure if my friend's cat has FIP? And do cats with FIP still shed the FCOV virus? 3 weeks after exposure, should I still worry about this virus potentially being active on my shoes?
There isn't any reliable test for FIP because it is a mutant virus. If you test for corona, you'll get a positive result in most cats, but that doesn't mean it is positive for FIP. You should not have to worry about the corona virus infecting your cat. Most adult cats have been exposed. I'd have to look up how long the virus stays infective, but it is not a very stable virus. Proper disinfecting of your clothes and shoes should suffice if you are worried.
If so many cats are exposed to the virus, how is it that so few actually develop FIP?
Because it is only in a small percentage that the virus mutates and causes FIP. It is very rare, luckily.