remove axle nut.
push axle though? you don't have to take off the drop out lock to get the axle out.
+1
Further, put a 1x4 board and shims under the wheel to take the weight off the axle while you drive it out. It may be corroded a bit and reluctant to move. But with no weight on it it should punch out easily. Certainly once its out and you clean it up, you'll want to shim the wheel up to where the axle slides in easily, you'll have several things to line up.
if you want to fix the original problem, maybe a hacksaw and chisel to get the head off, slide the lowering block out of the way, then you can get to it with a drill and penetrant.Or maybe the drill to begin with. With the wheel out of the way you can get a run at it.
Someday you'll want to fix that. Can't tell in the picture, but maybe the bolt should have had a washer under the head?
Looking at it again, I'd go to ACE and get a chromed Allen Head with a flat washer. That would look sharp.