A micro welding torch should be a smaller version of a full size welding torch - identical apart from size. It is most important a user thoroughly familiarises himself or herself with the correct usage and safety precautions because failure to observe the correct procedure can result in extreme danger. A flashback can have potentially serious consequences...
A person who is thoroughly conversant with the procedure and safety will know how to connect the hoses to the welding torch and to the cylinders - if it is a quality unit, it is an intuitive process due to the fuel gas fittings being left hand thread and the oxygen normal or right hand thread and the hoses are or should be different colours with blue being the norm for oxygen and red or orange for the fuel gas.
One final word - never, ever, under any circumstances use a lubricant on the threads of the oxygen bottle or associated fittings...
It would seem the low current control side of the welder is working ok - you squeeze the trigger of the torch and the wire comes out and the gas flows and the the main contactor clunks?
I doubt running on a generator has anything to do with the problem and it was coincidence - if the generator voltage surge or spike was going to cause a problem it would be the control side, the sensitive electronics. The high current side is much more robust.
Welding current is supplied to the torch through a cable and is connected to the wire by the copper tip the welding wire is pushed through.
Don't worry about measuring the amps - measure the voltage through the circuit from mains voltage going into the big transformer, probably 20 - 40 volts AC from the transformer, being changed to DC and reaching the contactor and then leaving the contactor and being delivered to the torch when the trigger is operated.
If the big transformer is ok, the most common problem is with the torch assembly where the constant flexing in use often damages the pipe, wiring, main cable and welding wire liner - a lot is forced into that thick tube supplying the torch which is why professional welding machines tend to use the Eurotorch so a spare torch assembly can simply be plugged into the machine and the welding job completed.
It could be the circuit that supplies the current that closes the contactor. You need to check the switching circuit with an ohm meter, the contactor coil with the ohm meter and finally the voltage on the contactor coil..
Metallic inert gas or mig welding for short - unless using flux cored welding wire, an inert gas is an essential part of the welding process.
The torch and tube assembly usually consists of a welding cable to carry the welding current to the wire tip of the torch, a wire liner through which the welding wire is delivered and a nylon tube through which the gas is delivered and a return wire from the trigger. The whole is encased in a single heavy duty tube.
I suppose the nipple could be the gas inlet - why not try it and see...
Most mig welders use an electro-mechanical contactor to switch the welding current through to the torch and others use an electronic means.
The heavy current transformer, rectifier and chokes are robust and rarely fail. If welding a high duty cycle they are protected by a thermal breaker that cuts the power until the unit has cooled.
You need to check welding current is available from the transformer, through the rectifier and choke and if so the problem is likely with the contactor.
https://www.youtube.com/watch?v=_yTnfRS9B04So, for gasless MIG welding polarity (when using flux core wire), your MIG welding polarity settings should be set to direct current electrode negative (or DCEN). With DCEN, the negative terminal inside your machine is hooked up to your electrode (your MIG gun) while the positive terminal is hooked up to your ground.
https://makemoneywelding.com/how-to-set-your-mig-welding-polarity-settings/