Zona,
The amperage you set a breaker at has no bearing on what you plug into it--for example, if I plug a 7.5 amp grinder into the correct voltage outlet with a 50 amp breaker (probably not going to happen, but just say I found one), this would do absolutely no harm to anything on the grinder side of it. The breaker only allows the amperage that the machine can pull--it has up to whatever amperage available to do the work.
So, answer to question 1 is technically, yes, where you set the machine parameters does regulate amperage, but no, as far as this hurting the machine, it will not.
Question two is unknown to me. I would guess that the welder is probably designed to pull a certain amperage, and that is what the breaker needs to be set for. You are popping breakers because the welder wants to pull 18 amps or 20 amps and the breaker only ALLOWS 15. It sees the 3 or four amps (or however much more) as something that should not be allowed (such as an accidental short that could start a fire), and breaks the circuit. The machine should have ratings on a label on the back, just check them out and size accordingly. You can go right on or if you think you may get something that will pull a certain amperage in the future, oversize with the 30 (though I don't know of many 120's with 30 amp breakers)
As always, be careful when wiring. Good luck!
gls
1. As OSUtigger said, no it won't hurt the machine. The machine takes what it requires assuming you are applying the proper voltage. You plug lamps requiring 0.5A (60W) into 15A circuits all the time.
2. I assume from your first post you have it powered by either a 15A circuit or a 20A circuit with other loads running at the same time. The big problem people have with high current draw devices like your welder is undersized wire feeding the device for the length of run.
The input voltage your welder sees is at the back of the box not the voltage at your electric panel. Ideally, they are the same but if your supply wire, both within the wall and from the plug to the welder, are too light you will have a voltage drop along the wire and the welder will think its running on reduced voltage. This limits its power and depending on the design may cause it to draw excessive current/pop the breaker.
I assume that if you have a 15A circuit you have #14 wire so going to a 20A circuit with #12 wire will mean less voltage drop since the #12 has less resistance so you will get better performance since the welder sees an input voltage closer to the full 120V. This also assumes the original circuit was long enough to have appreciable voltage drop. So will going from #12 (20A) to #10 (30A) improve performance? Not as likely and only if, once again, the run from the panel to the welder is long. So if you want to put a longer cord from the welder to the plug going to #10 may be the thing to do and it won't hurt anything and you will know it is "good".
You can use a 30A breaker if all of your wire is #10 and you have 30A plugs. It would be cheaper to use a 20A breaker and your 20A plugs with the #10 wire if you go this route since the welder should only draw its max rated current (20A??).
The circuit breaker in the panel protects the wire in the wall from overheating, it is not intended to protect anything plugged into the receptacles.