Under the old C: drive, under Users\Administrator, I could see all the default Document folders. But according to my girlfriend, quite a few documents are missing.
Did you check C:\Users\Public? Everybody always forgets that one :-) Also, some programs save their user data under their program folder. You mentioned architecture. Old versions of drafting/engineering software are notorious for this.
I right-clicked the user profile folder and hit properties. I found that the total disk space used by this folder is much bigger than all the visible files and directories added up together.
This is normal. There is always a disagreement between the total size reported by Explorer and the sum of all the files. Just remember to have hidden/system files showing (they are hidden by default).
While copying the "Administrator" folder it is asking me the for permission to copy files with some weird username prefix.
This is also normal. The files are NTUSER.DAT and USRCLASS.DAT (along with their associated backup and log files, which have long and weird extensions.
I tried to boot the OS from the old hard drive by selecting the USB boot option from my BIOS. The OS starts with Windows 10, but still I could not see all the files that mentioned by my girlfriend under C:\Users
You probably have Windows 10 on the machine you connected it to. Windows 10 cannot boot from USB by default. There are registry hacks that must be made.
To find the missing files, find out what extension they have and do a search for them. But like I said, check C:\Users\Public and the program's own install folder.