When I flip the switch to turn on the ceiling fan it works...for a while. Then the circuits in the room blow. Is the issue electrical or an old fan?
I can't say for sure what the issue was. I did go ahead and replace the fan with a new one based on the logic that the fan light can stay on for nearly ever, which made me think the issue was the fan motor itself.

I still don't have any answers on the light bulbs going out fast. I have a small hope that maybe the bad fan was causing some kind of imbalance in the house, but I have no idea if that makes sense or not from a technical perspective. 

In either case, the new fan seems to be working quite well and so do the 2 outlet switches - one for the light, one for the fan.