We've gone from CC6 to 7 to 8. No real issues that have not been resolved. The big jump from 6-7 changes the database from Sybase to Cache, so if you have queries going directly to the database, you will have to recreate them. The canned reports all work just fine.
Contact center really does not have anything to do with Telephony Manager, but you should be on TM release 4 to support release 6 on the pbx.