>>Can I safely use a propane burner indoors? Should I just start making 2 gallon batches until I can find a better place to brew instead?

Uh ... hell no! Plus, I'm sure your land lord will be less than thrilled
I suggest:
use two 2.5 gallon pots on your stove, spitting your ingredients, then place them in the sink to cool (I have a double sink), else you can put one in teh kitchen sink and on ein the bath tub. Then dump them both in your 5 gallon bucket, add the needed water to bring you to 5 gallons (I boil some the night before and after its colled put it in the fridge). This will help cool down your wort. Then place your bucket on the flor and rock it hard for a minute. Even in an apartmnet, you can handle 5 gallons.
But not a big 5 gallon stock pot, unless you have some super fancy stove.