No. Campaign missions (as based on the open beta) are fairly tightly defined by the map and checkpoints; if you revisit a mission, you’re basically playing it again.
You start at the same level each time you play either campaign, PvE, or PvP, and choose one of two skills each time you level up. It’s only binary choices each time, so not really a big deal.
Gear goes in load-outs, and you can select one load-out during character select. This is based on character/command level which are persistent stats INDEPENDENT of the in-game level of your helix (i.e. skills). It’s very different in this respect to BL games.
It doesn’t work that way. You have one profile, with a Command Rank. As you play, you gain Character ranks. At certain points, you unlock additional characters up to the initial 25 available (and apparently 5 more later on.) You only have one instance of each character. This will make more sense if you play the game!
I imagine that you will either not get grouped in a PvE or PvP match with players who lack a map, or the map will simply not be an option. That’s my guess based on the way the game type and map selection screens came up in the beta, anyway.
The couple of times I went off the map, it was insta-death. In multiplayer, my character got repositioned in “revive me” stance on the nearest edge inside the map. There were accounts of players finding themselves in odd locations, but there wasn’t anything in the beta to deal with that. I think the intention is to avoid such things happening, and patch where necessary.

