faulty gas solenoid? / dirt inside the gas solenoid?
IIRC the gas valve / solenoid should be inside the torch head on those..might be worth checking, could be the previous owner has dropped it and knocked something loose, or left it somewhere damp and some gunge is clogging it up?
I think you may have several faults on this machine (some maybe from previous owner trying to fix!)
First I would say you have a gas feed issue ref the "bubbling" As the gas valve is in the torch handle there may be an issue within the valve (bit of muck?) or a holed pipe that shows when the torch is moved (when you are welding!) If yo have access to a low pressure air supply you could try feeding low pressure air through while welder is unplugged but pulling trigger (this will save using up your gas) Could be leaking round swan neck. What ever it's drawing in air with the gas.
Second you have a connection/electrical fault. Looking at your pic 3 post 16 The first thick cable, down and to the left of the motor looks a bit cooked on it's sleeving, As this is a high current carrying cable it's not unusual to see some cooking but it does indicate that where it connects to may be a poor contact.
In the same pic what does the relay contacts under the plastic cover look like? (you may see SOME soot on the cover but over time the contacts can burn and so the relay operates but the power does not get across the contacts (can be intermittent).
From your description of the basic test you have done on the diode pack (metal finned device) they seam ok. But if you look at your pic just at the bottom of pic there are some spade connectors that you should pull off and clean and check there crimping. You could also try bypassing the cut out device (blue thing to the right of you screw driver) if this is acting up it may drop out the welding power mid weld. A made up double ended male spade connection lead would do this for testing.
Have you tried the welding wire from your new machine in the old one just to rule that out from any issues.
faulty gas solenoid? / dirt inside the gas solenoid?
IIRC the gas valve / solenoid should be inside the torch head on those..might be worth checking, could be the previous owner has dropped it and knocked something loose, or left it somewhere damp and some gunge is clogging it up?