Hi! Welcome to the Hammerwatch Community!
Yes you can be killed 'permanently' in single player mode, but when you step on a checkpoint it saves the game, allowing you to reload from that point. At the top of the screen to the center-right there should be a golden Ankh with a number next to it, that is the amount of lives you have left.
The presents are for a Steam achievement in which you need to collect 24 of them to acquire.
There aren't many settings besides the old arcard style ones and basic video settings, it's all about learning and conquering your obstacles (With secrets along the way).
Goodluck!!