Yep they last through death and everything. They also last for 8 hours! Not only that, but the different strength potions will also stack. So you could have a +100% a 50% and a 25% and use them all together to give a +175% xp boost. And there are items like the bag of knowledge which gives you a permanent xp boost as well, and those types of items stack with the potions as well.
As for the lockboxes, you can get by just fine without anything from those, but they make life much easier if you start opening them. Basically there are random rewards inside, from sucky to raid type items. You can get any type of xp potion, various mounts, and some insanely powerful items. But most common is junk potions like an evac potion and run speed potion and some even less useful. If you unlock them regularly, you will most likely end up with bags full of crappy potions. But maybe some xp potions, and if you're lucky... a mount or red item. But it's total gambling so you can spend a lot if you don't get lucky.