I found it easier to undo the other end of the HT lead, I left the wire connection intact at the pcb and just removed the small screw holding the electrode bracket so I had the pcb with HT lead and electrode bracket hanging from it. To do this of course you have to remove the casing.
The connectors however will come off, but as you have found they are tight on.you may have to apply some leverage with a screwdriver blade.
Regarding the soldering, I didn't do mine but I think you only need to remove the solder when you are removing components. I would suggest though that you apply flux. If you do want to remove solder then you will need some braided wick or a solder pump.
I suggest you examine with a magnyfing glass, without one the joints didn't look that bad.
The problem joints all seemed to be where the terminal posts were fitted.
Hope all goes well!