How to remove heavily rusted hinge
The usual tool to use for this is a screw extractor.  



You drill a hole in the screw, then use the screw extractor bit in the hole you drilled.  The brand pictured has the drill and extractor on the same bit.  

Drilling the hole is usually the hard part.  It may help to drill the smallest possible hole first, then re-drill with a bigger bit, then extract the screw.  

If you're patient, you can use a Dremel tool with a cutoff wheel or tiny grinding wheel to cut the heads off, remove the hinge, then use vice grips to unscrew the bits left behind.  

Rather than cutting a slot with the Dremel, you can try to just grind a flat on the head so you can get a better grip on it with vice grips, but in this particular case it looks like there's not enough room for that to work well.