Nope. Dying is the only way to "Game Over." You don't have to close any gates except those that are essential to progress the main quest, such as the one just outside Kvatch.
To use a Soul Gem, if it's empty, you have to obtain a Soul Trap spell (or a weapon enchanted with Soul Trap), cast it on a monster, then kill them. If you have an empty soul gem in your inventory, the monsters soul will go into the gem. You can then use the filled Soul Gem to enchant your weapons and armor.