Hi I have an apartment that is a 1 Bedroom on the first floor with other apartments above. The apartment is about 510 square feet... The only area that is in its own separate room is the bedroom. Would a 12k BTU air conditioner cool the whole apartment, including the bedroom? Without cooling it too fast, which would not give the air conditioner time to dehumidify the air.

Too large of an air conditioner is just as bad as too little... but that size of air conditioner is good for the whole apartment no problem...
Most site recommend around a 12k for 600sq.ft as long as you don't have direct sunlight all day long or constantly cook... if so then you would step up to a higher 1

Good choice on the 12

Yeah that's what I read is that if its too large then that won't give it time to dehumidify the air, but if its too small then it will run constantly.

I wanted to make sure that the 12kbtu would be able to cool the whole apt. including the bedroom, but not too big...

Here's an easy, simple BTU Calculator, perfect for your situation: Air Conditioner BTU Calculator And if you're worried, go up 1-2,000 BTU.