I would recommend we do a bit more testing here to be sure. You'll need access to a second Mac which you can connect to your system. We'll use Target Disk Mode to see if we can gain access to the systems HD. Here's a good write up to explain things: Use target disk mode to share files between two Mac computers. Once you get your system up in Target Disk Mode using the other Mac you should have access to your systems HD. Use the other systems Disk Utility to see if the drive has problems if it does then that is more likely the issue here. You still might not see the HD from the other Mac if the drive has fully failed.
If you can't get the system to boot up in Target Disk Mode at all then thats more likely a logic board failure.
As to your direct questions:
Apple uses a basic logic board within a series of systems. As an example your system only has one system within the series iMac14,4. So generally one can't use the logic board number alone to know what it is. The better direction is to use the Apple part number which you have here. This should be a direct replacement of what you currently have:
You might want to look at getting the board that supports both a HD and blade SSD
That way you can add in custom blade SSD to gain some real performance!