Think you are really going to be disappointed with the $150 machine.... The thing about welding is the machine is never big enough to do the projects that keep getting bigger and bigger..... Found that out when I got Hobart Handler 140 ($700) and now long for a Hobart 210MVP ($1200) because...
IF all the above items fail to show the trouble, you could pull drive motor assembly and test it with a 12V battery, most MIG welders use a drive motor the can range from 9 to 18 volts dc....
Sounds like a parasitic load ... IF you disconnect a terminal on battery and put a voltmeter on disconnected terminal and battery post and get a reading meaning you have a parasitic load draining battery..... Could be from slight load leaking through from charging circuit..... Simple solution...
THE pressure on return line to filter and cooler has no relationship as to height/location as to outlet of control valve or height of inlet to tank .... IT will either have pressure or not, pressure is caused by resistance to flow (in the sense-Plumbing) ...There has to be "some" pressure to...
Looks to me like a standard hydraulic system breather.....
https://www.surpluscenter.com/Brands/Dynamic/3-8-NPT-Chrome-Plated-Screw-On-Air-Tank-Breather-Dynamic-Fluid-Components-DB40-06-9-16088-6.axd
Cleanliness is next to godliness.... Also prevent lots of problems... Clean up fittings ... Check for crud between ball/poppet and base, and use caps/plugs to prevent dirt entering QD....