Sorry, I have no idea what those loose boards might be. I Googled the numbers and labels on them and didn't find anything useful.
If the ends of the wires were cut off rather than desoldered, they could also appear to have been never attached. Take a really good look around the interior and check for anything that looks like they may have been attached to and then snipped off. They appear to be ground wires, so it may not be on the circuit boards. They could have been attached somewhere else in the chassis.
I doubt anyone will be able to identify those cards or the loose connector unless they have access to the service and repair documentation for Sony receivers. You could search around and possibly find a copy of the 5300ES service manual and drawings, but you often have to pay for those.
I don't think you are liable to damage anything by powering it up with those cards not installed. I'd put the loose cable back in, too. It could have just popped off during transport or something, especially if the unit was not well packed. It happens.
You may have better luck asking for help in the thread devoted to the 5300ES. Just about everyone on AVS with one of those is probably subscribed to that thread and posting there will be more likely to get their attention. It's at
http://www.avsforum.com/avs-vb/showthread.php?t=880352