The ATP CTP graduation certificate is a requirement to take the ATM written per 61.35 (2). So yes, you have to complete the course before you take the written.
The ATP CTP has nothing to do with the written or checkride. The sims you do for the CTP will not relate to your checkride unless you happen to be taking the checkride in the same type aircraft the CTP used. So yes, you still have to take
an ATP checkride.
The ATM written test is valid for 60 months per 61.39 (a)(1)(ii).
If you fail the written test you must meet the requirements of 61.49. In addition AC 61-138 says the instructor who signs you off for the retest must be an ATP CTP instructor. No where do the regs require you to redo the ATP CTP.