But the Windows 10 setup and repair media is FREE for you to make USB or DVD copies. Google Microsoft Media Creation Kit to find out how to do this.

As to the drive, https://www.youtube.com/watch?v=HCuc0EXeCkg shows it's a lot of work but yes, once you get the drive out ( skip to https://www.youtube.com/watch?v=HCuc0EXeCkg&feature=youtu.be&t=455 to see the drive).

With the drive out you can put it in some USB case or adapter ( example: https://www.amazon.com/dp/B0787MDDS9 ) then fish out the files you forgot to backup.

This isn't a trivial job so your choice to have it done at your shop.