No, it's not that restrictive. Basically, what you do is (in most VA's) you book a flight from the system (any flight, just choose one you like to fly) that you're allowed to fly based on your hours flown for that airline. When you're relatively new, you'd probably be flying ATR's or Embraers, and progressing through the ranks you'd go to 737's, A320's and eventually on to the 747's, A340's, that sort of stuff. Anyway, you just pick a flight whenever you feel like flying.
Some VA's only allow you to fly on VATSIM, IVAO or Virtual Skies, or some other online network, but most VA's are OK with just flying it on your own computer. That really depends on what VA you decide to fly for. For offline flights, the VA is really pretty much counting on your honesty in submitting Pilot Reports, in which you fill in some details about the flight.
There was a list of virtual airlines on Avsim, but they got hacked so now I don't know where to find one. If you want a list of some good VA's that might interest you, without me advertising anything on the forums, send me a PM and I'll see if I can help you.