L2 - I'm not sure how I'd fix that if it were broken, but I think it's ok.

R19 is fine.

C15 is original, I have no way of testing it and no way of replacing it short of wrecking a known working board to donate the part.

I don't have one of those multimeters that can check capacitance/inductance - any other things that are critical to the horizontal sweep?

P1 does indeed have a scorch mark on the solder-side, as well as browned connection on the component side. What's to be done about this? Presumably something shorted or sparked... a design flaw? If I fix it, is there a preventative measure to ensure it doesn't happen again?