Do you have a check engine light? Even if you don’t it’s still worth it to swing past an auto parts store like Autozone or Napa and have them do a code scan for you. They’re free and only take a couple minutes. It’s possible your engine may be throwing codes without triggering your engine light. If they do find any codes though, come back and post them here and that will be a lot more helpful in targeting the problem.
But just based on your symptoms I’d guess it’s something in the evap system. Most likely a small leak or your purge valve is stuck. This won’t really hurt the car. It’s completely drivable with an evap leak, the worst that happens is that your MPG will suffer a slight decrease. That and you have to deal with the occasional smell of gas outside the vehicle.
This is all assuming you live in a state that doesn’t have smog inspections though. If your state does they won’t pass you unless you fix the problem. And in older cars like this it’s usually not worth investing the couple grand to fix a problem that doesn’t affect drivability since the cost of repair is usually more than the Valis of the vehicle. But again that’s just an assumption based on the most likely scenario, I could be completely wrong so take what I say with a grain of salt.
You can as long as it’s a manual transmission. But push starting isn’t exactly easy and really requires a second set of hands to do safely. It’s also not advisable to do repeatedly as it can damage your engine. It’s mainly meant to be used as a one time emergency fix, not a long term solution.
Your best bet is to either find alternative transportation while waiting on a new starter or putting the old starter back in, giving it a few good whacks and hoping for the best.