Also, is it worth taking the bike into a shop to get the wheels trued and checked over? I've done some work on tuning the derailleurs and such and am pretty comfortable there, but it's more the condition of the wheels, rims, spokes, etc that I'd be concerned over... a few years ago my wife was riding the bike while we were on a trail and ended up with a flat. She kept riding it for at least a quarter of a mile before realizing it. And possibly longer. Not sure if that would have done a number on the rim. FWIW, I have been commuting on this bike no problem and had the same tube that I changed back then up until a couple weeks ago when I noticed it had started going flat as well. So while it seems the back wheel is OK, I'm still a bit paranoid and am wondering if I should have it checked (or, if it's easy to DIY check, how to go about that).
If the bike had fat tires and she was riding on a trail, it's unlikely the rim was damaged, although it could have been if she rode over a big rock (but she'd have noticed that). If there was damage to the braking surface, you'd notice a pulsing feeling when you squeezed the brakes.
Remove the tire and tube. Go around the wheel and inspect the edges of the rim--where the tires hook onto--for dents, dings, and chips. Inspect the braking surfaces as well.
Wheels need to be trued only if they are out of true. Put the wheel, without the tire, back on the bike and spin it. Does it seem to wobble from side to side? Do the brake pads rub on one spot? Does it seem to have a hitch or a jump at any point in the revolution?
For flats, I carry one or two spare tubes. If I get a flat on the road, I change tubes, put the punctured one in my pocket, and patch it when I get home. If I get a second flat, I can always patch one of them on the side of the road.
If you put Slime in the tire, be prepared for a big mess when you do finally take the tube out.