Can you cast an aura without a target?

Aura spells (i.e. Cards cast from your hand, and added to the stack), do require a target. It is unfortunate that they are the only spells that target, without "target" within the rules text. Note the full text, though: "legal object or player according to the Aura's enchant ability and any other applicable effects".

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Do auras entering the battlefield target?

Most often, auras enter play after being cast. When they are cast, a target is chosen at that time. Then when the aura spell resolves, the aura enters the battlefield attached to the targeted permanent. Can you play an aura with no creatures? You must be able to select targets for a spell in order to cast it, so you can't cast an enchant creature if there are no creatures on the battlefield. into its owner's graveyard instead of entering the battlefield.

Do enchantments stay on the field when a player dies?

Nope. When a player is eliminated all of their spells, abilities ,permanents and all are removed from the battle field. even if they have a spell on the stack and are killed while it had not resolved, it is just exiled and thus removed from the stack. And another question, who controls enchant player? The player who played the enchantment controls it, no matter what it is attached to.

Consequently, how does totem armor work?

Totem armor doesn't destroy the Aura; rather, it changes the effects of the spell or ability. On the other hand, if a spell or ability deals lethal damage to a creature enchanted with an Aura that has totem armor, the game rules regarding lethal damage cause the Aura to be destroyed, not that spell or ability.