C37 sounds like one of the mains filter capacitors. Guessing by the number, I would say it's probably 0.56uF 400V 85 degree C rating, but the value is not critical and the replacement you found ought to work fine.

Transistors are fairly easy to test if you have a multimeter with a diode check function. Power transistors usually fail completely shorted between all three leads. I've seen some that got so hot they desoldered themselves and fell out yet still worked fine. Unless there is a crater in the housing, a visual test is not reliable for determining the health of a semiconductor.