If there's a cracked solder joint, start with resoldering the pin connector to the system board. Not all games use the same pins so that could affect which games work better than others.
Did you inspect all of the system board? Including the bottom by taking it out of the case? Maybe there's corroded traces on the bottom while the top still looks fine.
This. I would reflow parts slowly and test each one by one. Check any resistors, diodes, capacitors, etc. Reflow the pins under the cart slot.
Do you have a test unit Genesis? You could remove the ICs and test them on another unit to see what happens.