#Windows 10 error code 0xc00000e full

The most important, of course, are your own user data files (docs, etc.) and hopefully your backup of those isn't too far out of date. Depending on the age of the backup image, there may be a few things (user preference settings, updated applications, etc.) that will need to be 'caught up' manually. Restore first and then boot the restored Windows OS to see what, if anything, is missing. In fact, the BCD store, similar to a registry hive, is just a data file like any other that Windows loads during its start-up process.

#Windows 10 error code 0xc00000e full

In answer to your question, yes, restoring a full system backup image should restore everything contained in that image, including the BCD store, to exactly the same status as existed at the time the backup 'snapshot' was created. As you are now seeing those errors regardless of how you try to boot, either restoring a full system backup image or reinstalling the Windows OS, although an unwelcome solution, probably imposes the least waste of time in the long run. Unfortunately, the ultimate conclusion reached by most people when Windows repair tools fail to correct the problem is the same one you have arrived at. I strongly suspect that's a result of the OS update's changes to your drive layout rather than anything to do with Grub as such. Both errors that you are seeing (0xc000000e and 0xc000000f) are Windows boot errors meaning essentially the same thing: the Windows Boot Manager is unable to access some resource as specified in its Boot Configuration Data (BCD) store.