Pulled the Mess that was my Timing Chain....

IGNITION: Yes that is really, really, really the wrong ballast. Get a BWD19 as noted; it is pretty close to having the right designed-for resistance value, both hot and cold. I have made detailed measurements on several ballasts and yours is not right for a Mopar of this vintage. If you have a meter with a ohms function (resistance), measure this ballast cold; if it measures around 2 ohms like I think it will, then it is wrong. The proper cold resistance is 0.5 to 0.6 ohms, cold. A BWD19 only costs a few $$...put it in, and get that issue behind you; it may not solve the immediate issue but will cause poor cold idle at the least. (Don't be shy to speak up if you need help making this measurement if you want to check.)

BTW, the wrong ballast can cause the idle to go to crap after 10-30 seconds of idling, particularly right after cold start up.....does that sound like your issue?

On your spark testing, was it done like described, across a 1/4" gap? Testing by just putting the spark plug on the ends of the wires and seeing if the spark jumps that small gap is not a good test at all. The compressed fuel/air mixture takes a LOT more spark voltage to jump that .035" gap than it does to jump it in open air. That is why you have to test spark across a 1/4" gap in open air: so that it will be know to be adequate inside the cylinder under actual firing conditions. (I emphasize this since you have not described how you tested spark, and if not done right, you still don't know if you have good spark.)

BTW, I think you mean 9-11 VOLTS on the coil when bumping the starter? That sounds good and means that the ballast bypass function of the ignition switch is right.

Can you post a pix of the inside of the damaged distributor cap? I would encourage you to post this info on your www.slantsix.org thread; the guys there know the rotors/cap issues inside and out.

COMPRESSION: When you speak of the distributor always stopping in the same spot, that is odd; it should stop randomly when any cylinder is starting the compression stroke. (It might favor some cylinders but one just one all the time is abnormal.) It implies that just one cylinder is sealing up well. When you set the valve lash, are you sure the cylinder being adjusted is not anywhere near TDC on the overlap? I don't want to insult you; it is just one explanation of why the engine would always stop with the rotor as just one spot. Have you turned the engine over by hand and found resistance every 1/3 turn of the crank? Have you run a compression test? That would answer this issue and settle it; A COMPRESSION TEST AT THIS POINT IS NEEDED, IMO.

Another reason for poor sealing in these engines is sticking valves. I can't recall if you had the head reworked or not. If not, then valves sticking will prevent good cylinder sealing (and perhaps bad lash adjustments in some circumstances).

CARB/FUEL:
The occasional hole in the carb base was a mystery to me, and I thought it was just someone's hacking, but it was explained on a thread (by SlantSix Dan, I think) that sometimes these are drilled to get the idle or transitions mixture right (as I recall). It seems to be needed on just specific individual carbs.

The carb mist sounds good; the slight vacuum leak at the carb base is of concern. Have you sprayed something like WD40 or carb cleaner around the carb base while idling to see if that changes things? Have you put a vacuum gauge on the intake manifold under the throttle (true manifold vacuum) to see what you have? A vacuum test may not be useful with such a bad idle, so is probably only meaningful in those short spurts when you get it run well.

Disconnecting the fuel line removes the residual pressure that should stay in the fuel line for quite a while. (Residual pressure shows that the fuel pump's outlet check valve is good.) Why that helps? I can only guess at this point.. perhaps the float setting is high and this line disconnection allows the fuel level in the bowl to drop to where the carb runs right until the fuel pumps pushes more fuel and overfills the bowl again. OR the new needle and seat is crap. When you disconnect the line does any fuel flow back out of the carb inlet when you pull the line away? (That is a long shot check....)

Bad fuel? Try pulling fuel from a container rather than the tank. But, if the fuel was bad, it would not likely ever run right in any circumstance, not even when you have disconnected the fuel line for a while. So bad fuel does not seem likely at this point.

BTW, did you ever answer SlantSixDan's query as to if this was a '72 California car? He would know how that effects things better than anyone I know.