<<<release a version of Audacity >>>
Audacity is a complete slave of the computer that's running it.
<<<Three people singing with other instruments also being fed into the software>>>
But you're not, are you? There's a computer in the way. Although the software will cheerfully handle 16 tracks of audio in both directions, it's generally agreed that there's no way to actually get 16 tracks in or out because of hardware limitations. The computer is in the way.
So based on normal hardware available:
1) <<<Record someone singing whilst the computer is playing backing music >>>
There is a Preference setting in I/O called "Play other tracks while you record a new one." If the computer is up to it (that computer again), it will play existing tracks while you record a new vocal or guitar. And you can keep stacking them like that up to 16. This preference may appear in a different place depending on your computer and version.
Playing other, different software might be possible, but there you may run into Audacity liking being the only sound program running. Audacity is a really simple program and likes running on really simple computers.
2) <<<Record someone singing whilst at the same time playing guitar?>>>
Plug your microphone and microphone amplifier into "Left" and the guitar into "Right". They're both mono feeds, so this is valid. Plug that into the Line-In of your computer and you will get a recording of both at the same time you can split up and remix as you wish.
3) <<<Three people singing with other instruments also being fed into the software at the same time.>>>
That's an external mixer. You might be able to force a computer to do this (which I doubt), but actually using it would be very awkward and painful. No premixing, no headphones, no panning. You'd be missing so many performance mix tools that everybody would give up and go home.
I posted a general all-call a while ago about wanting to get all 16 tracks into Audacity at the same time. My checkbook is open. What do I buy? I never got a response.
There are much worse computer limitations. If you're on a Windows PC, there's a very good chance you only have a Microphone-In. There's very little you can do with that other than plug a microphone in.
Many people go on to external USB mixers to get production abilities.viewtopic.php?f=27&t=9477