sound of door with rusty hinges
It makes a rustling noise.
something may be rusty
As for your suggestions, the first one is incorrect use of grammar but the second is okay. You could also say Would you please shut the door? The would you mind part is really not needed, and can sound redundant or a little insincere. Would you mind really means something like Do you object. Someone earlier suggested "Would you please mind shutting the door", which really means "Would you please object to shutting the door." Sometimes 'mind' can mean pay attention to, and syntax should make the difference clear.
Creak
yes
creaking
A bus door typically makes a "whoosh" sound as it opens and closes. The sound is produced by the hydraulics and air pressure mechanism that operates the door.
Trevor dropped the door knob and he is having trouble shutting the door.
Because when you slam the door the door makes vibrations and sounds are made by vibrations.
slam
To make your door soundproof, you can add weatherstripping around the edges of the door to create a tight seal, install a door sweep at the bottom to block sound from coming in, and use soundproofing materials like acoustic panels or foam on the door surface.
It's most likely a degenerative joint disease and you should see your doctor.