sound of door with rusty hinges
It makes a rustling noise.
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.
something may be rusty
Trevor dropped the door knob and he is having trouble shutting the door.
yes
Creak
sqweek
creaking
Because when you slam the door the door makes vibrations and sounds are made by vibrations.
slam
It's most likely a degenerative joint disease and you should see your doctor.
Touching a blade will not automatically make it go rusty.