Also check for activity on pins 39 & 40 of UB13 (CRTC chip) as advised by Sirius earlier.
With regard to the PSU connector (J9) someone has soldered the wires directly because of poor contacts between the plug and socket (not an unheard of problem!). At some point you’ll want to come up with a better solution because the wires will have to be disconnected every time you need to remove the main board. For the moment just check that the correct voltages are appearing on the main board near to J9.
I wouldn’t be too worried about the slight corrosion on the IC pins although sometimes this sort of deterioration can cause chips to fail. In the end it’s a question of identifying faulty ICs if indeed there are any.
Alan