Can I enter the "restricted area" early in the game?

I generally like to work on side quests before the main quest. So far I've started a couple (Bane's and Zsasz's quests, for instance) that have objectives inside a big red "restricted area" on the map. I don't know what happens if I try to cross this boundary, as so far I haven't had reason to enter it.

Is it feasible to try to completely finish these side quests early in the story? Or should I wait until I have access to this "restricted area" later in the game? If I can't finish, is there any benefit to partially completing them now?

As far as I know, it's not even possible to enter that until later in the game. There's a giant wall completely surrounding it.

Side quests don't give any rewards except trophies(except the first AR training mission or unless that's why he's doing it) and with the exception of one trophy, there's nothing that you can't do after beating the game.

You can't enter the restricted area at all. The titan containers that you are referring to are most likely in the subway area that runs beneath it. You haven't unlocked the map of the subway yet but when you do you will understand. You can also go inside wonder city, but you can never freeroam in the restricted area.