I think for now we should concentrate on the motherboard because the PSU is such a mess we need to get the soldering right before we attempt repairing it.

On the motherboard pin3 does look better but still not as good as I would like. Also the flux residue and other debris is a concern, although zooming in on your pics makes things look worse than they probably are. Having said that we cannot afford for any of the debris to cause shorts - no more bangs!
See if you can clean things up using an old toothbrush and some alcohol based fluid, perhaps meths. You can buy PCB cleaner but if you have something suitable we can save time and money.
Alcohol is also usable as a flux, some fluxes are over 90% alcohol, so perhaps you can try to use it instead of the flux to see if you can get the solder to flow better. If you compare the pins you have not soldered you will see how much neater they are with the solder flowed to the bottom of the pin and no stray bits left further up the pin. Pins 1 to 7 need attention and also the last 3 pins but concentrate on a few pins until you get good results and perhaps only add a small amount of solder.
If you have an old/scrap circuit board it may be useful to practice on that first, remove solder and re-solder. Good luck and let me know how you get on.