Ok, I am just transitioning to the E-2/C-2 community, so I don't have the "in the E-2 squadron" experience, but I've done 2 cruises in S-3's hanging out with these guys and am in the E2/C2 RAG now, so I'll answer as much as I can.
The E-2 guys do get plenty of flight time. More than the hornets, less than the helo's and S-3's. The difference is that the E-2 guys get some of their flight time as copilots so they don't get as much left seat time. This requires strong managment of flight time by Ops to keep everyone in night qual, etc.
As for LSO, there are lots of E-2 LSO's out there. In 2 cruises, 3 of 4 CAG Paddles were E-2 guys. Not necc'y the rule, but the opportunity is there. There has even been C-2 CAG paddles, but that is a difficult path for a COD guy.
The COD is the 3rd Carrier aircraft that I've flown in the Navy and it will be the most challenging of them all to land on the boat. With the T-45 and the S-3 both being jets, you forget how to use the rudders. If you don't use the rudders correctly in the E-2/C-2, you will be so far behind in controlling the aircraft, you will not get aboard. The aircraft requires true stick and rudder skills to fly. Once you get the rudders under control, now you need to focus on centerline. The 80' wingspan of the airplane is almost double that of a Hornet and Paddles will not tolerate you not being on centerline. No you are not taking these aircraft into the merge, but the challenge is trying to fly this aircraft well.
The E-2 pilots do not love staying on station during a mission, but they are also involved in the mission. They can and do assist the NFO's during high workload times. Once the mission is over, I have found more than 1 E-2 down at 200' doing some SSC until recovery. These guys do manage to have some fun.
As for being a jet VT instructor, yes the opportunity is there. There are several E-2 guys in Kingsville and Meridian.
Ok A4's, I'm interested to hear what you have to say about this as well!