Luke,

"Will I be able to plug the computer into the PA system that I have now and get it to play a backing track from Band in a Box...and at the same time will I be able to plug my guitar and my wireless mic in as well...."

It should work without any other peripherals....Maybe just a cable to run from
Laptop 1/8" output to aux input. Pretty standard. Assuming that your amp can simultaneously accept "aux in" and Guitar / Mic inputs.

If not, cheap mixer will do the trick for practicing... or this:

I looked at your Karaoke machine... I believe it has a usb input for Flash drives.
(I am sure that whats it for) You can render WAV or MP3 backing track on BIAB write it to thumb and plug into your machine and it should accept it, to play along with that backing/minus with your guitar/mic.