Yes, pulling on the cord will create pressure that can pop out the decomp, even if it does not fire up. The black fluid coming out the muffler is fuel mixed with carbon (probably from the exhaust port and muffler). I have an 034 I am working on that does this too. The saw seems to flood easily. Sometimes when pulling on flooded saws, you will see the unburnt fuel coming out the muffler in good amounts. Smoke is present when there is not clean combustion. This can happen especially when unflooding a saw. Usually when a saw is running, you don't see any smoke it produces because it is evacuating it at a whole lot faster pace. When the saw moves slower like when yours dies, then you will see any smoke it produces because it isn't blowing it away as fast.
The coil will run with the break in the casing, that is until crud gets in there and starts to corrode the wiring. If you have access to a soldering iron, you can tin the wires with solder and then wrap it in electrical tape. With a break like that, there might be more damage to the wire that you don't see. You might have some strands broken under the sheathing and when the wire heats up from the ignition pulses, it might be causing issues with the spark at that point.