I was getting a simmilar problem when I was in my last appartment and I found out that it was mostly due to older bananas. Still edible, but getting black spots on them.
Also, we were getting flies and spiders and later found out that our downstairs neighbor had tons of bugs, and there were even more in the basement. The place was infested.
I don't know of a good solution besides keeping fruit sealed, but maybe one of those fly zappers?? I don't know if fruit flys are attracted to them, but it should be fun to watch anyway.