Well, a reset issue is almost always either your power connector J101 (heat damage or bad solder joints on the male pins) or Bridge BR2 and cap C5. The bridge and cap are most often the problem. You can also get bad continuity between the bridge and cap too from cracked solder pads on each. If you change them, I'd suggest running a jumper on the back side of the board from the + of the bridge to the + of the cap and also from the - to - to ensure a good connection. At the minimum you'll need to test for good continuity with a meter from the + to + and - to -. You mentioned that you pondered replacing the bridge on the flipper board. The bridge on the flippertronics board has nothing to do with the 5v circuit, which is what runs the CPU. When it goes low, it causes resets. That's all on the driver board. A bad bridge and cap also won't cause a constant reset all the time if they are starting to fatigue and fail. As they begin to go bad, you'll sometimes get very sporadic resets and that will usually progress to a more constant reset and eventually can lead to the CPU just never booting at all.
The odd switch problems shouldn't be related to the reset issue. They are totally different circuits and voltages. The switch matrix gets it's 12v derived form the 18v lamp matrix. Sometimes a low 12v to the switch matrix can cause weird switch issues. Ribbon cables can as well. You might try reseating all of the ribbon cables to see if that clears up the switch issue.
I'd be curious though if all of the switches are working 100%, especially your optos. If you go into test mode switch edges and activate your switches by hand, do you get multiple switch closures on any switches, specifically the optos on row 2 especially since the fire button is on row 2? The optos are run by several LM339 chips on the 16 opto PCB under the playfield. When one or more of those chips goes bad, it can cause all kinds of wacky switch issues. I've had for example a game activate the correct switch, as well as a switch that isn't used because of a bad LM339. The right fire button (switch 12) is on the same row as the lower right opto (F2), which is the your flipper opto for the lower right flipper. Since that's the case, you probably should check that whole row out for a possible bad switch diode or multiple switch closures with the optos. Check the column 1 too for a broken switch diode. There are no optos in the column 1. Specifically I'd like to know if you hit the right flipper button in switch edge test mode, does the fire button also activate?