These Tiki cards and similar (like Shapesister) can be somewhat confusing.
The ruling on Trap Monsters indicates the following thing specific to this situation (emphasis mine):
... If "Book of Moon" is used against an already active Trap Monster, the Trap Monster is set as a Trap Card again and can be activated and Special Summoned the next turn.
So your Tiki Curse actually returns to your Spell/Trap card zone, set.
Other important details from that link include:
...and are subsequently treated as both Monster Cards and Trap Cards. As such, they take up both a Monster Zone and a Spell & Trap Zone at the same time.
This tells us that they take both a monster and spell/trap zone. Derivated from that fact we have the consequence that they can be destroyed both by effects that destroy monsters or spell/traps (from this other link).
While active, a Trap Monster can be destroyed by effects that affect Monsters and/or Spells/Traps. (You can use either Dark Hole or Heavy Storm to destroy them.)
Finally, if they are negated, they get caught in some sort of "limbo", where they return to their Trap zone:
If they are negated with "Jinzo" or "Royal Decree", they will return to the Spell & Trap Zone as Continuous Trap Cards with no effect, and will remain there even after "Jinzo" or "Royal Decree" are removed.
Some further reference on Trap Monsters on this Konami blog.