A single external hard drive isn’t enough for backing up files and folders. There could be data corruption. I only fix these issues using a Mac. You should take the drive out of the enclosure if you can and put it in a dock.
It’s not worth $800 to replace the logic board. You might as well get a new Mac. The Rossmann Group can replace the RAM chips on the logic board since they are soldered to the logic board. It’s also possible that it isn’t the RAM chips and something else is happening.
If you’re talking about your Windows user account password, you can reset it by putting Hiren’s BootCD PE on a USB Flash Drive or DVD. Instructions to copy the software to a USB Flash Drive is located on the “USB Booting” page. If you go to the “HOWTOs” page, it will tell you how to reset the Windows password using the software.
Here are the following upgrade options for your Mac: https://eshop.macsales.com/upgrades/imac-27-inch-late-2012-3.4-ghz/InternalDrives? It would be easier to replace the 3.5" inch HDD with a 2.5" SSD. Taking apart the iMac is a hassle already. You shouldn't take it apart if you aren't a professional.