First off buy a quality #1 phillips screwdriver. Some of the better ones have serrated tips to help hold the screw. Second before you try turning the screw put the screwdriver into the screw & hit the end of it with a hammer driving it into the screw. This will help set it into the screw tightly & the shock/harmonics will help to break the screw lose. As for the stripped screws try a small visegrip & the screwdriver to turn it. Again hitting it to shock the srcrew!
I'd say your on the right track looking at the carb it's probably gummed up especially if it's had ethanol fuel standing in it for some time. Clean the carb pay attention to the holes in the cross section/the upper tube of the pilot jet/the smaller of the jets. The ethanol turns to sugar & will plug things up. Use the smooth part of a tip cleaner, small wire, or pick. to poke through the holes to clear all of them. You don't want to use the serrated part of a tip cleaner, or a drill bit, as this will open the size of the hole & change the jetting. If you can drain the old gas & put some fresh ethanol free premium in it & use only ethanol free gas if possible. Good luck
