Wow, that's a difficult problem, BN. Sounds like you should get your attorney involved, as well as an independent engineer to measure the severity of the noise. You might want to investigate replacing the garage door entirely, perhaps with a different kind of mechanism.

If the noise is as bad as the rent-stabilized tenant claims, then I would suspect that a rent reduction will be awarded. As a consequence, the sponsor will demand a reduction in maintenance for that apartment and will probably get it.

As for the slamming of car doors, your Proprietary Lease or House Rules (or both) probably contain language along the lines of, "The Lessee shall not permit or suffer any unreasonable noises or anything which will interfere with the rights of other lessees or unreasonably annoy them" (a quote from our Proprietary Lease). Unfortunately, your lease probably doesn't give you the authority to levy fines for such violations. Your sole recourse would be a Pullman-style eviction, which is such a Draconian response to door-slamming that it probably wouldn't be supported.

(I need to emphasize that I'm not a lawyer, just a Board President who tries to stay up-to-date on issues affecting co-ops.)