The computer "sound card" (I prefer the term "audio interface", as sound card usually infers only a specific class of interface) is important for determining the quality of the audio going in and out of the PC. In other words, audio being recorded to the computer has to go though the sound card first, and when you play it back to listen to it, that playback signal is sent though you sound card to your speakers.
The sound card, however, has no direct effect on the quality of your mixing on the PC, other than how it may affect the sound of your speaker playback. The quality of sound mixing is determined mostly by the software itself - both the mixing software and the plug-in software.
And yes, with a few rare exceptions, all the mixing work is done by the computer itself. The main difference computer configuration (CPU speed, amount of memory, etc.) makes is that it'll determine how much horsepower you have to mix with, which in turn will determine how many audio tracks your software can run at once with how many simultaneous plugins per track. The faster and larger you computer config, the more of each.
There are two main exceptions to the computer doing all the work. First are those plug-ins that take advantage of seperate dedicated processor cards to do their work. Examples would include the plugs from UA and Sony Oxford. These tend to be the most high end expensive plugs (usually costing more than the computer itself these days.) The second would be those plug-in sound cards that have their own MIDI DSP (Digital Signal Processor) chips on them for performing MIDI track processing.
HTH,
G.
