What kind of curtain rod should I buy for these brackets?
Those brackets are for a 'pull down blind' aka 'roller shade', not for curtains. You would pull down the opaque blind and it would internally catch. A short tug to release it and it would recoil itself. You can see these in cartoons when they abruptly spin up and fwap-fwap-fwap at the top position. Roller Shade bracket 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Outside Mount Roller Shade Bracket

If you are determined to put curtains on the door(s) then chip off the many layers of paint covering the screws and cut through the paint around the base before attempting removal. A heat gun or powerful hair dryer will speed this process up but be careful about overuse unless you plan to ultimately strip the door down to bare wood and refinish.

The curtain rod you purchase may or may not come with a pair of appropriate curtain rod brackets. There are a wide variety available for different applications. For a small curtain rod meant to accommodate the window in a door, there's a good chance the existing mounting screw holes will be reusable.