Inspiron 7559 new M.2 SSD install OS from backup

The factory image backup you're referring to is intended to allow you to restore the factory image onto the SAME hard drive where the backup exists.  There isn't an easy way to use that to restore onto a brand new hard drive.

However, do you actually WANT to do a clean install, or would you prefer to just migrate your system exactly as it is right now onto the new SSD?  If the latter, you could simply use a tool to clone your current drive straight to the new SSD, then remove the existing drive, and then you're done.  Your entire system will be on the SSD just the way it is right now.  Macrium Reflect Free can do this for you.  You should be able to run the clone from within Windows itself, or you can build the Rescue Media it recommends and boot from that in order to run the clone.

If you WANT to do a clean install, you can download Windows 10 directly from Microsoft here.  The file you download will allow you to create CD/DVD or USB installation media for Windows 10, and you would then boot from that in order to perform a clean install.  This however will be a generic Windows 10 installation, not the exact configuration that this system had from the factory, which may be a good or bad thing from your perspective.  However, you will at least need to reinstall your drivers when starting from a generic installation.  The easiest way to do that is to install "Dell Command | Update", which will download and install the needed drivers.  Then if you purchased any software directly from Dell, also install "Dell Digital Delivery" to redownload that.  You don't NEED to keep a backup of that since you can always get it again later, but of course you can if you want to.

In terms of whether formatting the HDD will erase the system image backup, it's stored on a separate partition, so if you only formatted your other partitions, then it would be preserved, but if you ever reinitialized the entire disk, it would be lost.