How to know if there is damp or mold behind wall?
The only 'test' the inside of the wall is to actual visually inspect it and to acquire access to do that would require 'damaging' the wall. The wall can be repaired but I'm assuming that you don't want to do that due to your lease and the cost.

There is a way that might be allowed under your lease and is relatively inexpensive and easy to repair. This is the use of an inspection camera which is a device that uses a skinny tube with a camera on the end, much like the doctor uses to look down a patients throat or for a colonoscopy. They can be rented for the day. To use, a small hole is drilled through the wall covering, and that hole might be smaller than the maximum allowed by your lease for general use like hanging pictures and shelves. Even if not, it is small enough that it could be repaired fairly easy though due to the many different types of wall coverings I consider that a different question and answer