[chumbucket]

One of the biggest problems that I see you're facing is that the wiring is so old and such a mess, it's going to be difficult for anyone, unless truly experienced in boat wiring, to help out. The pics are a bit fuzzy too. I honestly can't help you with the color codes. Boaters World and West Marine both sell an excellent book on 12 volt boat wiring that goes into detail about common color codes that were used in the marine industry. I'd pick up a copy of that if it were my problem. A couple of observations that I might be able to help with though;
The mystery part appears to be a hydraulic cylinder and the boat perhaps had hydraulic steering at one time. I wouldn't worry about that. It's not part of your current problems.
Those type of solenoids are pretty generic and can also be purchased at BW or West for relatively short $$. I do believe they are the source of your problems. A solenoid is nothing more than an electro-magnetic switch that when 12 volts DC is applied to the coil (the small wires), the electro-magnetic coil is energized and pulls up a post that completes the load circuit (large wires) supplying the juice to whatever is to be actuated. I would replace both solenoids.
And, I would try connecting the yellow wire, bypassing the plug after the new solenoids are replaced and see what happens.
I would also be sure your neutral safety switch is working correctly in the control box(just another thought of a possible problem).