The Save mechanics in Borderlands 2 - save stations that record progress (and provide you a respawn point on death) combined with fast travel points that double as starting locations when you load into the game - are a result of having a game that is both a single player game and a multiplayer game. It's a compromise; you lose a little bit from the single player side (the traditional "save and load exactly where I left off") so that multiplayer isn't super weird.
The mechanics at work here are not new mechanics, though perhaps you have not experienced them before. Diablo 2 most famously did it, as does Diablo 3.
My advice would be:
Where possible try to tackle one quest at a time and don't start anything new if you think you won't have time to finish.
Run past monsters/through areas that hold no interest to you. Borderlands 2 is a game that suggests perhaps you should stop to see every sight and do every little thing. But you don't need to do this.
I believe there are a number of quests you could accomplish in 15 minutes, but some of them would require you to skip everything of interest on the way there and back (as it were). Some set up such that you can turn around and start doing them and then turn them in rapidly.
But some of them are involved, and it's ideal if you can carve out longer play-blocks to deal with them. This is especially true of story quests, so be aware.