For starters, I hope the command that you ran,
cat /dev/sda > windows7.img didn't try to save that .img file on the
Second, what was the reason for doing the whole disk, rather than the Windows 7 partition, i.e.
/dev/sda1, or whatever it is?
Third, if you have a disk large enough to hold the whole image, why not just install Windows 10 on that disk and boot it, and save the Windows 7 disk, unused, until you're sure that you're either keeping the computer, and Windows 10, or rolling back to Windows 7? Remove the new disk, replace the old disk, and it's a Window 7 machine again.
Now the question. Windows 7, as installed, has a license, and Key that should be on the new sticker that the refurbisher attached. That key code will always be good for that computer with Windows 7, no matter how many different times you install other versions of Windows on it. If you have the install DVD for it, or if there is a built-in way to create one, then you can even wipe the disk and install from scratch and use that same Key code. Of course, doing that will loose all the custom drivers, and other setup, done by the refurbisher. If you do install Windows 7 multiple times, eventually, when you try to activate it, you will be bumped up to a live person who will verify that you have only one installation with that Key, on only one computer. Still, they will activate it.
If you do as planned: upgrade to Windows 10 (if the free upgrade actually is still available), then that should include a new Key code for Windows 10. Things may have changed as of Windows 8, or higher, so I'm not sure, but in the earlier versions, once you have that Key code, if you later wiped the disk and then did an upgrade fresh (requiring, of course that you install something to upgrade from), you could, again, use that same Key code for the upgrade. Always, by license, and by their system, limited to exactly one installation on one computer. So, the Windows 10 installation will have its own Key code, and license, neither of which invalidates the Windows 7 Key code or license. If, after the upgrade, you blow out the Windows 10 and restore the image you made, it will be as if you reset time. Unless you change some hardware in between then and now, it won't even need to be activated again. As far as MS is concerned, nothing ever happened.
As mentioned in comments by @Run5k below, the free upgrade expired almost 7 months ago, so the process may be moot for you. Free, or not, the details are still the same. So if you buy and install Windows 10 rather than the free upgrade, the situation remains as above. The benefit to a purchased install DVD is that it's not tied to the copy of Windows 7 installed on the machine already, so if you revert the computer to Windows 7 and give it away, the Windows 10 is still yours to install on any other computer you get later.
As to hardware changes and reactivation, there is a guarded algorithm that Microsoft uses. The installation process examines the computer and compiles a "signature" for it. Different elements carry different weights in that process. When Windows reboots it re-examines the hardware and creates a new "signature" and compares that with the one it has from installation. Once the two differ beyond a certain threshold, reactivation is forced. This is, BTW, a totally offline process, so going offline will not defeat it. Since it's a guarded process, I can't say how much you can change before it triggers, but the more integral it is to the computer the more likely it will count. Adding a sound card likely won't, neithe will changing the video card, or adding more drives. OTOH, doing all three might do it. In Windows XP I know that changing the NIC, and nothing else, was enough to cross the line, while adding a second NIC was not. If you do other upgrades to the computer, then revert to Windows 7, it is possible that they may add up enough to require reactivation, but if it does, it's a simple process. It's usually automated, and even when it's an extreme case, once you get to an operator on the phone, they have you verify that you have one installation, on one computer, and it's reactivated. At one point in '07 I was doing some expirements and multiple reinstalls of XP and Vista, and ended up doing phone reactivation 3 or 4 times a week. But, every time, it was the same thing; just explain that I'd wiped and reinstalled, or that I'd made major system changes, and activation code was given.