I got the edition with all the dlc not too long ago, and when I played, after the first few missions, icons for the 3 dlc campains appeared, and I could do those at any time. I think it was after I got to the hideout in the southern part of the map. I didn't care too much for the mech dlc, but the sea base one gets you a really cool boat at the start, and at the end, you get a lightning gun with infinite ammo (it recharges itself). The jetpack wingsuit and the lightning gun make the normal game fairly easy. You can take out bases by flying up, popping your parachute, then frying everything from the air.

I believe you would have to complete the DLC again to get access to the rocket wingsuit, I started again and I don't think I had immediate access to the DLC content I think I had to progress through some of the story mode to unlock it, I can't remember exactly tho it was a while ago but it definitely won't be already available for use without completing the DLC for it.

Sky fortress is a DLC that is separate from the base game. You won't have to restart the campaign to get the add ons. You should get a mission prompt and get the Bavarium Wingsuit in that mission. It's a pretty great DLC and I handgun with it