Yes, you can use FBS (or NBCS) as a block which might work better than BSA. If you are having issues with high background, this is worth a try. Some people use casein regularly as a blocking agent in WB/ELISA and it can be much superior to BSA.
The other thing you could try, if still having problems, is to block with the serum of the species your secondary antibody was raised in. I remember trying that for WB once and it helped clean up non-specific bands so I don't see why it shouldn't work in blocking for ELISA too. Hope that helps, and I'm sure you'll get some other useful answers...
I don't see why not - heat inactivated serum will be partially denatured so if anything it might work better as a block. But I've never compared them, so maybe someone else will be able to comment from their experience?