Congrats on the first victory. Some general tips:
- If you go for a master trait, stick with it. It's a lot better to focus your specialization even further than try to branch out in an attempt to make up for flaws. In the case of MAD, for instance, taking that third level of TaN and going for Ironman (so that 100%/200% are even more health than before) will keep you alive while you pelt your attacks.
- Shotty runs, for the most part, consist of two typical scenarios: shooting at monsters up in your face, or shooting blindly into the distance to check for said monsters. Shotgun-type weapons automatically hit, but their damage drops off per tile (between you and the enemy). Best to worst, drop-off rates are: combat shotgun, shotgun, and double shotgun. The opposite is true of bullet spread, though, so if they're close enough, double shotty will hit just about everything.
- Shotgun-type damage (known as shrapnel) takes twice as much damage to penetrate armor when compared to physical. This is why the master trait Army of the Dead is so useful: all enemies' armor is reduced to zero automatically. This makes long-distance attacks that much more effective, especially considering drop-off rates.
Yeah, the Vaults take some time to crack. The middle vault requires a little more creativity than the others, though.
Next time DEFINITELY put the agility mod on your boots, not your double shotgun. Seriously, it's worthless on a gun that requires no accuracy to hit things.
On a personal note, I highly suggest trying to win Hell Arena just about every time you can (certain challenges and Nightmare! aside). It's a great source of experience early on, and there are some useful rewards should you clear it entirely.