1) It should
2) DoTs are their own standalone DoT damage. If it's anything like Demolish, only the initial strike will deal additional pierce damage provided Disembowel is up. If there's no initial strike then Disembowl won't affect the DoT
3) Try to keep CT up as much as you can, but really you don't have a choice in the matter. You have to keep Heavy Thrust and Disembowel up as much as possible (preferably 100% uptime) and you'll be doing CT anyways to finish the chain. If you keep Disembowel up 100% of the time, there should be no reason why CT isn't the same