I think it's part of a bug, not being able to form a team once your inside the summon area.

However, you can form a team outside the summon area (in the group base, at the frog npc Fukasaku) and then enter the summon area. You're better off this way, actually. That way, more people can get the buffs that are dropped from fighting the enemy ninjas in the first phase. Once that is done, keep your team together and go fight the snake/summon (you can now do this fight as a team). If you end up going through all 10 rounds before the snake is dead, then you'll get attacked by the elite ninja, but in a team so hopefully the battle is easier. The elite ninja bit is still a major waste of time though. I've defeated the elite ninja in a team and nothing happened. Also, at the end, there is an anbu npc where you can drop items (or something, I'm not entirely sure, haha) that you apparently gather during the instance. Once your group gathers enough of each type, your group gets a buff (for the next group summon capture, I believe).

Basically, have a buddy/trio system ready before going in, and stay with your buddies till the very end.

Update! My group had a summon capture tonight, and I did defeat one of the hostile elite ninja squads with a team (I didn't even notice that there were three before; there's an Ay team, an Onoki team and a Mei team), and this time I noticed something dropped for defeating them. 1 Spirit (which is one of the 3 buffs you can get during the first phase), 50k coins and 5 coupons. It was... quite underwhelming, haha. But it's something, I guess? It also looks like the damage you do to the hostile ninjas counts towards your total damage. I was late to the summon capture, missed the first phase and didn't even reach the snake to fight it (since I got ambushed lol). so all I did was the elite hostile ninja fight, but I still got 15 coupons, exp, coins and a summon frag for it.

Also, note that in your team, whoever is in the center is mostly likely not gonna get to do much against the elite hostile teams. The middle team will be soaking up the damage and debuffs, mostly. xD

Yes, you can fight the snake in a team. You just need to form the team before entering the summon capture area. ^^

That's what I experienced when I was in the center against the Mei and Ay teams (I haven't encountered the Onoki team yet, but I can assume the same). In their own teams, the kages seem to have infinite initiative so they'll go first. Mei immediately combo'd my first move and controlled it so I couldn't do much with my main damage dealer. The middle team also eats all of the standards, so it kinda makes sense. My third and fourth position ninjas were the only ones that got to do any damage, really.