I know the low tire pressure warning will come in in my parents 2013 Highlander if the spare tire is low. We couldn't figure out why it said low tire after we checked all of the tires and they were fine. On a whim, we checked the spare and it was low. Turns out it has a TPMS sensor in it as well. Put air in it, and it was fine ever since.
As far as the door ajar and interior lights staying on, sounds like a door latch issue or something. Is the tailgate fully latched/secure?