Can I lose more life than I have?
Yes, you can have a life total below zero. This is mainly a consequence of rule 107.1b:

Most of the time, the Magic game uses only positive numbers and zero. You can’t choose a negative number, deal negative damage, gain negative life, and so on. However, it’s possible for a game value, such as a creature’s power, to be less than zero. If a calculation or comparison needs to use a negative value, it does so. If a calculation that would determine the result of an effect yields a negative number, zero is used instead, unless that effect doubles or sets to a specific value a player’s life total or a creature’s power and/or toughness.

A player's life total is a game value, so it is possible for it to be less than zero. In addition, the last sentence specifically calls out that it is possible to set a player's life total to a negative value or double a negative life total.