"What will this mean for the wave propagation?"

Depends on the excited TMxm,n,p or TExm,n,p mode. Some modes produce currents along the periphery of end plates, while other modes do not.

"Should I better isolate the cavity wall from the piston, or should it somehow contact the cavity on all sides?"

I asked over NSF, and rfmwguy answered the best thing to do: 

"Contact, no gaps (especially at 25 GHz)"

SeeShells plans to use a conductive galvanic seal gasket between the plate and sidewalls.

X_Ray added: "While moving the plate the S-Parameter signal will look a little noisy that depends on the contact between the frustum and the plate(will be stable again after movement). The resonant frequency could be a little bit lower than calculated for a given length (MHz range, caused by longer current paths) if the metal seal is at the outer side. For TE01p that is not the case, no noise while moving the plate."

BTW X_Ray is a professional German microwave engineer and proposed you contact him, first by PM at NSF forum, so he can help you on that matter:

http://forum.nasaspaceflight.com/index.php?topic=38203.msg1419318#msg1419318

Thank you very much for your help. I will also contact X-Ray soon.
Jo proposed to make an endplate with a threaded endwall, so there will be always electrical contact to the cavity walls. He will machine this one in the next days.