There are a couple of ways to do this. You can display a different set of lyrics for every patch right in MainStage by adding a text box in layout mode, in edit mode you can type(or copy/paste) your lyrics into the box in the bottom inspector window. These will show up within your text box in perform mode.
Another option is to use a lyrics or charts app on an iPad (or several iPads) that accepts midi in and out. When you change the patch in MainStage your iPad (or all iPads) will change to that song with the lyrics, or you can choose a song on the iPad app and it will change MainStage to the appropriate patch. If you have several group members requiring the same lyrics you can set one iPad to control midi out and all the other iPads to accept the incoming midi program change so one person handles the changes.
Now..... as for your backing tracks, I'm not sure if my suggestion is exactly what you're looking for. I only have experience in changing patches with midi but not with backing tracks, though your problem interests me. Is each backing track in MainStage set on a separate patch or are you able to set it up this way? What is your process for loading new backing tracks? I've learned in the last several months that when using midi, the possibilities are practically limitless and depends much on your imagination.
If you can describe your setup a little further and how you have things arranged in MainStage I'd be happy to make more detailed suggestions you could try.