You provided a good data point for this issue: 9.4 volts to the 2 blue wires on the ballast, with the ignition switch in RUN (not cranking), is too low; it should be no more than 1/2 volt below the battery voltage. Get out the wiring diagram and start to trace from that point back to the ignition switch with the switch in RUN, and then back to the battery; you have excessive voltage drop somewhere. That needs to be fixed as you have a bad connection or wire or ignition switch.
BTW, in your last pix, the ballast has a crack in it. Replace it as it could be causing the 'off like a light switch' symptoms; that's not obsessing! LOL
You gotta go into the steering column and get to all the connections.....no easy way that I know of!
I have to wonder if that cracked ballast was the cause of the sudden cut outs... it would if the crack extended into the resistance wire inside.
As a test, you can temporarily jumper a wire from the large lug on the starter relay to the blue wires to the ballast and see how that improves things. But ultimately you need to fix the voltage drop; it won't get better and it will effect charging system regulation too.
The firewall contacts are commonly corroded and so the drop there is not at all unusual. So, find some Packard 56 terminals and a good crimper for them; resist the temptation to just kluge in something. You probably have others in the same condition that need replacing. The ones carrying the alternator current into the dash area and also making the connection for the power to come in are very susceptible to burning up, as these carry by far the highest currents. The MAD bypass you may have rad about is all about bypassing those.
For some of the firewall contacts that are not corroded, a good cleaning will help. Apply some dielectric grease around the contacts when you are done and ready to assemble it.
As for the drop in the ignition switch, the current flowing through the blue wire is causing the voltage drop in the IGN1 position of the ignition switch. Basically, the plating is worn off of the ignition switch contacts internally, and there is resistance built up there. So, you should also go ahead and replace the ignition switch for sure. That is one likely culprit in the cut-out issue; a worn out switch is common. Easy to get at NAPA.
All the above is a 'pay me now or pay me later' proposition. So ignoring it is not a good idea.
As for the small can, based on the pink and yellow wire colors, look for an 'accessory connector and time delay' block on your FSM schematic. It think it is a relay for putting power onto things like...well actually I don't know what things! Look on the label to see if it says 'relay' anywhere. If it is, then it will be warm whenever the relay is active.
What is going on is there is no current through the IGN1 position of the ignition switch when nothing is connected. So there will be no drop at that contact or its connectors and the drop to 12.2 volts inside the ignition switch is due to other drops. Once you make the connection, then the current draw on that line causes added voltage drop in the IGN1 contact inside the switch and that shows up in the lower voltage. In fact the extra current will cause added voltage drop at ANY poor contact or connections between the battery and that point. This is because voltage drop in wires, switches, connections, etc., usually ONLY occurs when there is current flow. If you want to isolate the drop location precisely, keep the connection connected. BTW, it may be tempting to not replace the ignition switch, but IMHO you are just causing more extended pain for yourself....each connection and wire and switch contact from the battery to the coil+ is a suspect until proven otherwise.The drop im seeing isn't at the firewall its at the connector right under the steering column. I'm going to try and clean that thing up today because it is literally dropping a volt and a half once that connector is connected(12.2 by itself and 10.5 after its connected). I will be looking at everything else on that connector to see if maybe something else may be causing the drop but it just doesn't seem normal to me to drop 1.5 volts at a connection.
I do have a question though. What is the standard reading at the coil with the ignition in the on position? I'm getting right at 4.8volts at the positive connection on the coil.