what is the best way to drop out a threaded screw?

So I've got some rusty old floor screws that I can't loosen with a Phillips head (they're old, so I'm betting it's not posidrv). The notches are all rusted and it hardly bites at all.

If I drill them out, I know I'm in for a hard time but am I better off going for a really fast rotation or slower?

And should I start with a narrow drill bit and work my way up (historically I've broken a few doing this in the past) or just begin at the intended diameter?

Yes, in fact I do work with computers - isn't it obvious?