What type of weather stripping for this door?
I'd suggest you remove the old and put in some nice new stuff. 

To remove the existing screws, first scratch any paint out of them (possibly with a sturdy knife). Then, using the right size slot screwdriver, you'll be able to remove them. (Firm pressure while turning is often what you need. Failing that, tap the back of the screwdriver with a hammer while turning. Failing that, get out the vise grips and just grab the head and twist.)