Some soldering irons have a floating tip so it is possible to solder with the power on (or use a cordless unit) - in any event I would never solder with the power on more so to the maybe some melted solder falling someplace bad or the solder your are holding inadvertently touching something as well.

That being said with regards to the problems I would start with bad solder connections - check all the connectors. Also if any of the connector are a darker brown you would need to check the pins on those and make sure they are making a good connection. Also wouldn't hurt to re-seat all socket IC chips (power off - unplugged even better before doing this). I would also re-seat all the connectors as well. Pins over time oxidize and could create connection errors. By simply unplugging and plugging back in is usually enough to scrap through the layer.

Next check the boards are all properly grounded - there should be some test points on the boards that you can measure from to the ground braid for continuity (if not can always measure from some of the IC's).