When I click serial keys it is written Serial Keys Depleted.
No more serial keys for Stalker Clear Sky?
When will there be more?

I guess serial key(s) are needed only for multiplayer or am I wrong?

What about the multiplayer update?
Should I install it only for single player mode or is it purely for multiplayer?

I think you won't be needing a serial key to play the multiplayer since there's already a thread in the STALKER Clear Sky forum that says in order to play the multiplayer you need to download the multiplayer update. You can find the download for the update at the same place where you download the STALKER Clear Sky (it's located at the bottom part). I may not be sure whether it would work or not on this one. I own the game but I have yet to try it. Hope this helps!

P.S. I also experience the same problem.

EDIT:

Sorry I didn't read your post thoroughly. I think it's worth a shot to try downloading the multiplayer update. I mean you can still reinstall the game if the update actually messed the game up and didn't work.