1 - Dead Shot is a full round action which means it takes all of your actions for that round. The only other action available is a five foot step. Part of the Dead Shot ability, however, is still rolling for your other allowed shots, even though you are only taking one. Each success adds to the damage of the single Dead Shot. I'd suggest rereading the description for Dead Shot to make sure that you understand it.
2 - Since using the double-barreled musket to fire both barrels is a standard action, you should be able to fire it while using Vital Strike and taking the -4 penalty to the attack roll. Your GM may interpret it differently and not allow this...
You added to your question there. Lol
I recently had to run a few gunslingers using Dead Shot, so I've had this come up in play recently.
3 - With Dead Shot, every successful hit (beyond the first) that your attack rolls produce adds another die to your damage pool. In your example of three rolls for the level 11 gunslinger, if all three hit, your base damage will be 1d10 (for the starting damage) + 1d10, +1d10 for a total of 3d10 before other damage bonuses.
If one or more of your attack rolls during Dead Shot are critical, then you roll once to confirm them all at once. You roll to confirm using the -5 penalty (which reduces itself by 1 for each critical threat beyond the first). If you confirm one or more, it's a critical hit.
Critical hits double the damage dice, so in this case, you double the dice you're adding to your damage pool for Dead Shot. If you confirm one roll, then your base damage will be 4d10 +1d10 +1d10. If you confirm all three rolls, then you'll have 4d10+4d10+4d10.
I hope that helps.