Old man with old car, new here with questions

Will a BBS carb for the 225 Cu in engine work on the 170 Cu in?

Yes, a Holley 1920 or Carter BBS carb intended for a 225 will work just fine on the 170 (…is part of the reason why I pointed you at the particular, specific carb I pointed you at). The factory started using the larger ("225-sized") carb on the 170 in 1967, which raised the 170's published horsepower from 101 to 115.

The temp gauge works and the fuel gauge will move a little and then go back to empty.

OK, if the engine temp gauge works correctly but the fuel gauge doesn't, that's probably a faulty sender or sunk sender float, but could also just be a rusted sender ground clip—that's a metal strap that bridges from the sender outlet pipe to the metal fuel line running to the front of the car; the system's supposed to ground through the fuel line-to-body attachment, but sounds like your car is very rusty underneath so there may not be an adequate ground. You may need to make and attach a new ground wire to the sender plate and run the other end of the wire to a good ground. Repair parts, see here.

The good news is that the tank is full and so far not leaking.

That is very good news; the '60-'61 tanks are nigh on impossible to find.