Yes, you can install the old hard disk in the new PC . And yes, you can access all data on it, only the drive letter changes. For some you have to take ownership.
And no, most software on that old drive will not run. If it runs, you can even copy the folder from Program Files (x86) and Program Files to your new c:-drive and delete it from the old drive. If it doesn't run, you have to install it in your new Windows like you did in the old Windows and, of course, you can delete it from the old drive (since it doesn't run anyway).
If the PC runs the new Windows on the new drive, it's safe to delete the Windows folder and the Program Files and Program Files (x86) from the old drive, although you might need to boot into some Linux distro to circumvent security.
So I'm building a new pc with all new parts (including MOBO), but I still want to be able to use my current HDD's files, since it contains MS Office certification-bound software and some other files. I'd be freshly installing windows 10 onto this computer (my current windows is bought under a licensed retail edition). I'm also buying a new separate HDD and SSD, and I plan to install windows onto the SSD for fastest results.
I'm confused with if I'd still be able to access files from the old HDD, and what I need to do with the old HDD since windows is currently installed on it. I also have access to an external hard drive if I would need to transfer files or something.