Electronic – Header plastic melts while desoldering

You may use an new IDE connector and attach it on the ribbon cable (I'm supposing they are using the standard size of wires). This way you can create a derivation without damaging the cable.

Another good practice is to use a buffer for all the pins. I know its a bit costly but otherwise the input impedance could cause interference in the system.

First of all: modern PCBs use lead-free solder. It has a higher melting point and a different "look" when solid. Sometimes it helps to apply a little bit of lead solder to get a better heat transfer between the soldering iron and the solder joint. After the whole soldering joint is molten, I use a vacuum plunger ("solder sucker") to remove bigger amounts of solder. Remove the soldering iron to avoid overheating the component.

Then (the component wire is still in the hole) I use some quality brand desoldering wick: I place the desoldering wick in such a way that it has a good contact to the solder joint, and then press on top of the wick. This way the wick has a good contact to the solder joint.

About heating up the capacitor: It may well be that you overheated the cap, I'm not sure if the metal case is has a good thermal coupling to the contacts.