I have a store and my database/server is installed on my [C:] Drive [Win7] but I don't have the installer anymore...
How do you access the database files? Do you have a specif program to retrieve and save data from the database?
I was planning to reformat my drive [Upgrade to Win10] because it's been running slow. It's not downloadable through the internet 'cause it was just passed on to me a few years ago. ...
Just by installing Windows 10 your computer won't magically run faster.
If that PC is not used to connect to the internet, you could just clone the disk into a faster SSD and keep Windows 7.
Is there any way to back it up without leaving any files at all? All my store records are in it and I can't afford to lose any data.
I assumed you have backups of your data if you "can't afford to lose any data." A slow disk could be a sign that the disk its close to the end of its life span.
In which format does the database files are saved?
I had an idea that to add an additional hard disk drive that is fresh [Windows10] but I'm not sure if I can use all the files of my old hard drive.
You could add a new disk and install Windows 10 on it, keeping your old disk with Windows 7 on a dual boot configuration. If going with that, make sure disconnect the Windows 7 disk while installing Windows 10 so it doesn't write anything on your old disk.