Is it normal for GFI test button to trip GFI when the circuit is dead?
Ok 2 parts, yes you are misunderstanding how a (listed) GFCI completes its test.
The test is completed by the internal circuitry creating an imbalance.
The clicking noise you hear is the mechanical debounce  that helps the reset function work better (not all GFCI’s use this method but many do because who would trust a micro switch smaller than a pencil eraser that you can not feel it click?
As for your welder if it is plug in with 2 parallel blades you only need a 15 amp receptacle having 2 receptacles it is legal on a 20 amp circuit (the feed through is rated for 20 amp). So it would not have been a code violation.
A 15 amp and 20 amp GFCI have the same circuitry the only difference is the shape of the receptacle itself.  They both trip at ~5ma and have 20 amp feed through if supplying other receptacles.