Remove the carpet.
Look for sources of moisture.
If a leak is not evident, it could be poor air circulation.
Are any of the closets walls on a outside wall?
If yes, there might not be adequate insulation in the wall which could cause a condensation problem.
Wash the walls/ceiling/floor down with a bleach solution and have a fan blowing in there to dry it out completely.
Address any issues you find (leaks, insulation lack, consider a louver door which will allow better air circulation, etc).
Paint the closet with a mold resistant paint (Watertite Mold & Mildew Proof Water Proofing Paint is at Home Depot).
Consider using some Damp Rid in the closet, and make sure you don't over stuff the closet in the future (to keep air flow circulating through it).
Many a closet has these problems and you can take care of it yourself.
Many old houses have creaky floors - it's a function of how the old joists and nails work (nails can loosen over time).
A few screws can fix some, but a lot of times a creaky floor will tell you when/where people are moving around in a house - think of it as a type of security system.
Not all mold is toxic.