ikeinblackriver, my thoughts to your two questions:

1. As I understand it, BIAB was designed more as a creating tool for backing music rather than a live performance tool. So there are basically two ways to use the program.

A. Use it live in the performance by an on stage computer.

Advantages: You get the full power of BIAB on stage including variations in the accompaniment, ability to change tempo and pitch, ability to repeat sections or songs,
select on the fly from thousands of songs on your computer, change instruments, mute instruments, and if you don't feel like playing for a while let the soloist do the work.
You can also have music notation or lyrics displayed on screen instead of carrying paper books.

Disadvantages: You must bring a decent computer to every gig (and watch it while on your brake ... ), it's difficult to try to do the things described above and also play, unless you have a "sound engineer" who sits at the computer and knows how to do all those things when needed. Also computers are known to crash or freeze without warning.

B. Use it at home to create backing track and songs to be played as audio files in the performance.

Advantages: You have all the time you need to prepare the best backing tracks you need, and modify them if needed. You can read the help files and watch tutorials.
Which you cannot do on stage. You can create different versions to a song, say long and short. After you create a backing song you can save it as an WAV or MP3 audio file.
You copy all your songs to an IPOD or another MP3 player, and use it in your gig. You can create play lists and folders and when you go on a brake just put the thing in your pocket
MP3 players are extremly reliable and sound very good. Most have built in very effective EQ

Disadvantages: You are working with fixed and static audio files - you have little control of pitch and tempo and zero control of the instruments in the audio file - every time you play that song, the backing track will always sound exactly the same, compared to using BIAB live that generates real changes whithin a song to sound much more realistic.

(there is a third way of using BIAB - exporting the backing tracks to MIDI files and playing the files onstage with a MIDI player that includes a sound module - there are a lot of advantages here but it opens a new subject that may not be relevant to your situation).

Personally, since I'm usualy alone on a gig with a guitar, and need both hands to play and fiddle with my mixer, amps and microphone, I use an old 20 GB IPOD connected to my mixer. I do save and copy my files to the IPOD as WAV files. The sound is very good, but the files are big, but with 20 GB there's no problem. I also use a $30 wireless remote control for the IPOD so I have real buttons instead of the finicky touch wheel.

Your second question: "what specific kinds of equipment do I need to get us set up to play in a piano lounge environment based on our instruments and using BiaB with a PC?"

I do not see you carrying the quad-core PC + 24" widescreen LCD to a gig - so you'll need to get a notebook computer if you want to use the A method. Or use the B method and bring a MP3 player with recorded tracks.

For a sound system, I have used and highly recommend the very portable "Passport" PA by Fender

Toronto, Ontario, Canada