Problem #1: Starting requires some kind of prime from the carburetor and yes it is choke related if it runs good all the rest of the time. The choke issue has a lot of carb'd Jeep owners going crazy. You may need to rebuild or clean the carb up. It should squirt some fuel in when you attempt to start it up.
Problem #2: Replace only whats necessary if you had no issues with the T-Case before. But you will have to see if you did more damage other than the stock tailcone. But what is more important WHY did the driveshaft break???
89YJ 2.5L TBI, AX-5, NP231 SYE, Tom Woods Shaft, 33" BFG T/As, 4" lift.
AF Jeep Club Tail #184