Battery draining fast

Yes the voltage regulator should be well grounded to the chassis as is everything else on these cars. ( Bold because you will do well to remember this. ) Dang near all the wires are positive wires. Ground path is a daisy chain of parts and hardware instead of a copper wire.
The original voltage regulators had some parts mounted on the outside of their underside. Todays replacements have modern electrical components that removed the need for those external bits. You would need to lift yours to see which type it is.
One of your pics showed a strange something poked out from under one side of the reg'. If that is shorting something or if that green wire that was poked under was shorting something the regulator may be dead.
Then sometimes the cars mounting holes for the regulator are striped. Some owners relocate the reg'. Other owners add a ground wire from their 1 good screw to a good ground connection nearby or all the way to the battery ground.