It is specified a couple of pages before the spell (Street Magic p161):
Note that spell effects may sometimes create modifiers that add on to existing modifiers from gear, implants, or adept powers. Unless stated otherwise, these effects are cumulative (though the gamemaster can always rule otherwise). Spell effects do not stack on top of other spell effects, however—only the strongest effect applies.
Note that even though spell effects don't stack, spells themselves do. So there is nothing preventing you from having a fire and ice aura at the same time. Another interesting point regarding Element Spells (Street Magic p162):
A spell may have more than one elemental effect—each effect adds a +2 Drain modifier, and the elemental effects are combined (though in some cases they may cancel out).
Then, applying it to your questions:
What happens if you have more than one type of aura on you at once? Do the DV bonuses stack?
Nope, as they are both spell effects.
Does it count as all the types of damage for purposes of resistance (so 2 fire resist and 2 cold resist would give you +4), or just the best, or just the worst?
This one is tricky. Technically, you have both spells at the same time. Your attacks can't be Fire and Ice at the same time, as that would require a "[Fire & Ice] Aura" with a +2 Drain. However, it is safe to assume that you could switch between Fire Attacks and Ice Attacks at will.
If you actually use a "[Fire & Ice] Aura" with a +2 Drain, the effects are combined, meaning you apply them both. Then, Insulation and Fire Resistance would both apply (better use different elements!).
What happens if the melee attack already has AP on it?
It stacks if the AP doesn't come from a spell effect.