I believe you have a standard 68mm BB shell, so if it were me I'd get a Shimano UN54 or equivalent in the same spindle length as your original (maybe something around 122mm?) measure the spindle from flat to flat (yours *might* have threaded posts for nutted style crank bolts, if so you *don't* measure the length including those).
Removing the fixed cup is what will require special tools (and a bit of expertise) and you'll need a tool to install the sealed BB unit, but you should be good for many more miles after that. If you had nuts instead of crank bolts, you'll need those bolts, too, but some BB units come with them. BTW, you need a crank tool to remove the cranks to begin with.
The deal with the chain falling between chainrings sounds like a FD adjustment is needed, if it's indexed then an adjustment to the cable housing tension...worst case is your chainring(s) may be worn or bent. If so you may find it cheaper to get an entire replacement crankset rather than 3 new chainrings, as there are some really low priced Taiwan units on the market nowadays. Cheaper isn't always better, tho.