No time limit is part of the game. The only thing it limits you is how long the story can occur for. You do not necessarily have to do the story AND get survivors or either for that matter. This game is very good in that sense as you aren't punished for meeting the requirements the first time through. You can still continue and try to level up to make it easier later on.
As for the map openness, most of the area is open at the start. There are things like shortcuts and paths such as the ramp to the underground tunnels on the strip next to the yucatan casino that is closed until midway through doing the cases. Most paths are unlocked by doing cases, which you can't fail any until that case to open that path. While none are essential they do make traversing all the areas easier.