Start with checking all your connections on the MPU and lamp driver boards, you'll no doubt find some dull, oxidized pins and connectors.

"BR" stands for bridge rectifier. To check for continuity of fuses, it's easiest to set your meter to 'beep' or the speaker setting between 200 and the diode setting. (->+)

The 'soot' under the flipper solenoid is probably carbon dust, the common dirt found around most solenoids that create problems as it finds it's way into mechanical operations. (linkage, bushings,sleeves & plungers, etc.)
The resister 'doo-dad' is actually a diode, yes it needs to be there. Make sure you get it placed the proper way when soldering. (note the band on one end)
I can't verify the wiring though, the pic isn't quite clear enough on my screen.

If a poor connection is the case, clean the fuse holder clips also, although they really should be replaced if it's a bad connection - that means they are weak and will pose more problems later.

If you're sure of good contact when checking the fuses and they are cutting in and out on the meter, just replace them.