Hello!
Well it could reallybe any of those things you listed... The heated core and thermostat would need to be changed out by someone that knows a car, if either is an issue. I would kind of rule out the core myself being that it really is not that old. Are you the original owner of the car? If so, then it should still have Dexcool in it, unless you had it flushed once.
For the coolant bottle, yes if it is low, it will be slow to warm up. I cannot comment on where it is under your hood, but mine is on the left side of the engine compartment. The bottle is somewhat clear and has line on it that should state COLD, WARM or LOW, ADD on the side. Open the hood, take a flashlight and look for it. Then look for these lines, add coolant if it states LOW or COLD, tkae it up to WARM or HOT. Then you will know it is full and can suck out a little more if needed to top off the system. The owners manual should show the bottle.
You can also remove the radiator cap. If you do this, make sure the engine is stone cold! You could get burned if you open it after it has been running. Remove the cap and check the fluid there.
