I set my D7100 to M (manual), set the aperture to f5.6 and the shutter to 1/250. After I take a picture the shutter speed changes. Is this normal?
No, it is not normal. One thing that Manual mode is used for is to take multiple photographs with the same exposure parameters which is very useful in many situations. There are a few cases where it may happen though. If you are bracketing for exposure, then one of the exposure-parameters, usually shutter-speed, is changed between frames until the bracket is entirely taken.
Some cameras have a feature called something like Exposure Safety which adjusts exposure on your behalf if the exposure you chose does not match the metered one. Check if you have that option in the setup menu and if it is disabled, should it be there.