If you have a 3 Ohm coil and run it with either the pink/white resistor wire or a ballast resistor the coil will operate at low voltage with reduced current flow.  That will produce a weak spark as you said.  

However, if you fit the remainder of the ballast wiring there will be a white/yellow wire from the solenoid to coil (+).  If that white/yellow wire is present and connected properly, the 3 Ohm coil and resistor won't make it any harder to start the car than it would be for a traditional non-ballast ignition.  Once started though the spark would be weak and could cause running problems.  If the white/yellow wire is not connected, then the engine could be very hard to start, particularly on a partially discharged battery or in cold weather.
 

I will answer all the questions later. Not yet finished testing and seeing what wants to work. 

If you have a resistor lead and a 3 ohm. coil what's the result. Weak spark cranking?  This is not related just a good place to drop it. Steve (CTR)

If the coil and its wiring were original to the '59 car they would be "standard" (3 Ohm coil without a ballast resistor or resistor wire).  The quick check on the coils is to disconnect all the low tension wires from them and put your multimeter across their low tension terminals.  3 to 4 Ohms = Standard coil used without ballast.  1 to 2 Ohms = ballast ignition coil that needs a ballast resistor or resistor wire.

The Pertronix obviously shouldn't have been making noises.  That suggests too much current flowing.  If that's the case it would explain both the overheated coil and the crackling Pertronix.  When you measure the resistance of the two coils let us know what you find and what wires are on each coil terminal (color codes and where they go to if possible).  

I realize there are different coils and that one takes a ballast resister. The coil is a Lucas and was mounted on the power unit when it arrived. I have rebuilt the power unit making many updates as I go. Being a 59 850 I'm sorry to say just about every repair made on the car's power unit over the years has been a dodge. Some people were just not meant to have tools.

I will check the two and see if they are different. The last time I had a coil burn me it had shorted inside and while the car would crank and idle it would not take fuel. Confused me at first seemed a fuel problem but again was a Lucas issue.

After bore, clean, hone, cam bearing and deck block. Change to and grind large tail crank, convert to diaphragm flywheel, and clutch assembly with two pc. Cooper balancer all balanced along with pistons and rods. Complete cyl head update to lead free seats and valves. Cyl. head surfaced .050 for a 9.4 C/R.

While messing around I converted to late tensioner type timing. Of course this requires a Cooper Rad. shroud and different front mounts to clear balancer. This was a running driving car when the engine was removed and sent to me. The dizzy a early DM 2 had a Pertronics unit installed. While not quite the correct one they did get it in there. A closer look showed the vacuum advance plate fouled by a screw holding the Pertronics in. To make matters worse the cent. advance weights were rusted to the pins. Sorting all that and the screw had damaged the vacuum plate and it would not rotate smoothly. Moved everything over to a 25 DM, once running the timing mark was jumping around. I shut it down and when I turned the ign. switch back on it sounded like beacon frying under the dizzy cap. Removed Pertronics and used points and condenser.

40 psi on starter, 50 psi running, 180 water, 2.5 psi fuel, 15.5 O 2.  Should make someones 59 mini very happy. Steve (CTR) 

Guess: test stand probably did not have a ballast resistor, coil wired directly to battery.  Second coil had higher primary resistance so ran cooler.  Steve, measure the resistance primary to output, compare the two coils.  Dont run the plastic rubbing block points with coils taking a lot of primary current, the plastic eventually melts.

Steve, any chance your first coil was for a ballast ignition and the second one was standard?  If you put the ballast coil in the wiring for a standard coil it will carry almost twice as much current and run very hot.

Running in a engine on my test stand I was having trouble rotating the dizzy as it was hitting the coil. When I went to move the coil it just about blistered my hand. I could touch it but could not hold on. The engine had just completed 20 minutes run time. I changed and relocated the coil just enough to move the dizzy a couple of degrees. After running about 10 minutes the coil I changed to was warm but I could hold my hand on it. How warm does your coil run? Steve (CTR)