You've gotten some good advice here, but some not so good. It's pretty jumbled up.
You have 2 issues to deal with. The welder, and the current wiring.
The simple answer is that the wiring is COMPLETELY UNNACCEPTABLE for a welder. Wonkothesane got the closest when he told you you need to run a heavier wire. Of course, you never told us the distance or the load of the welder you want to buy, so no one can tell you the wire size.
Now for the current wiring... you've got a fire waiting to happen. While I would agree with Ian F that it is not the worst I've ever seen, that is NOT a current panel box (it's 30+ years old), is NOT properly labeled (were does circuit #30 go?), and is NOT properly wired (circuit #28 an #30 are tied together). Jensenman got closest on this one when he described the bad contact inside the 220V breaker that wouldn't trip.
You've got a 220V breaker on (2) 110V circuits. It is NOT safe. The problem is that if one circuit faults, it will likely not be strong enough to trip both legs of the 220V breaker. this would leave both circuits hot and shorted- fire. The GFCI helps a little, assuming it is wired correctly, which is a BIG assumption. It is not (just my gut feeling, which is usually VERY accurate). I'd bet good money there is improper grounding all over the place, and likely aluminum wiring tied up in that panel.
Throw all your crazy ideas about running a welder off the two separate legs out the window, unless you are prepared to burn the house down. BTW- if you DO burn the house down, there is a good chance YOU could be liable, now that you've confessed a willingness to overload the circuits and operate equipment not suitable for the circuit.
You are ALREADY overloading the circuits with your compressor. That's why it's tripping. Stop trying to come up with a work-around and take the warning signs.
Stop guessing. Several people have already said that no further information can be given without pulling the panel cover and/or outlet. It's not "OK" because it's been grandfathered. It's unsafe.
Call the landlord to fix the wiring PRONTO. While a real electrician is there, talk to him about proper wiring for your welder.
Diagnosing electrical problems over the internet is about as smart as getting a cancer diagnosis and treatment at the local laundromat.