electricity questions?

I would think you are running everything on a single 20 amp breaker the way I understand it. and a 220 leg running to your compressor ?
I would pick up a small breaker box and run your 220 to it and install a 220 breaker for your compressor and a pair of 20 amp breakers in it.... That way when you use your welder it will be feed by the bigger wire service, just don't run the compressor at the same time. No, you can not and should not try and loud one side of a 220 breaker to receive 110 amp service.
and yes!! you could have a bad breaker that has been thrown to many times and throws at around 12 amp's and your welder is most likely drawing 15 amps or even 18 to 22 at demand. there is a 30 amp single , bur if you just have 12 ga. wire that could be asking for trouble. Get all your power into a breaker box you have feed to your shop now and install new breakers