I don't think it's possible to explain how this "ends up working" without getting really deep into the mathematical stuff.
The biggest difference is that a real photon can make a detector click, but a real one can't. The only difference I can think of in addition to that is that virtual photons don't have to satisfy [itex]E^2=p^2c^2+m^2c^4[/itex]. (This implies that their speeds don't have to be c).
All forces in all quantum field theories. There is a QFT of gravity (hence "graviton") but it isn't very useful since it lacks a nice mathematical property called renormalizability.
I could answer this with "none of them" as well as "all of them". Virtual particles show up in the mathematics when you expand a certain function in a series and consider each term separately. To say that virtual particles "exist" is equivalent to saying that the individual terms of that series describe what's "really happening", while the sum doesn't. I don't think there's any justification for that. Hence "none of them" is a reasonable answer to your question. However, they are a part of a method to calculate the probabilities of each possible result of any experiment, and these methods work extremely well. Every experiment that involves quantum mechanics in any way is evidence of that. Even the existence of stable atoms is evidence of that. Hence "all of them" is also a reasonable answer to your question.
Personally, I don't think of virtual particles as a description of what's really happening. I think of them as a part of the easiest way to do calculations.
When you do the calculations I talked about you're supposed to integrate over the momentum, so all speeds up to infinity are included.
Forget about the Dirac sea. It's an old model that's been replaced by quantum field theories.
The calculation method I mentioned also includes virtual particles popping in and out of existence in the vacuum. This is sometimes mentioned as an explanation for a non-zero density of vacuum. I'm not sure I like that explanation though. The result of the calculation is that the density of vaccum is infinite. If we try to guess a cutoff energy and only include smaller energies than the cutoff in the integrals, the result is still many orders of magnitude too high. If we take the cutoff to be the planck scale, the result is a ridiculous 120 orders of magnitude too large.