Hi Ben,
If you enter your Product Key on the sticker into that website you'll soon find out if its OEM or not.
I understand your concern about downloading the ISO of Windows 7 and it possibly being corrupted too though. Afterall, given all your issues its quite possible your Windows Update service is corrupted. There is a troubleshooter for that if you visit Microsoft's website:
http://windows.microsoft.com/en-us/windows-10/windows-update-troubleshooter
Could a friend or work colleague allow you to use their computer to download the ISO files? Could you go to a local library and use a computer there? So long as their computers are not having troubles that seems like a simple means to resolve this once and for all. I'm not sure about the Windows 7 ISO, but I've downloaded the Windows 10 ISO and know that was 4.11 GB in its compressed size, so all you'll need is a reasonable sized USB to copy the ISO over so you can transfer it to your computer later.
Many years ago a corruption to one of my systems running a 64-bit Windows Vista led me to eventually telephone Microsoft and have a chat with them. This led to me having a regular email contact with their technical support team. We spent a lot of time redoing some of the methods I had all ready tried to resolve the issue.
My ATT Z222 will not communicate with my computer and its all I have for imaging.
Eventually it became quite apparent to them this was just not going to work. At that point they asked me to take a photo of my Product Sticker and email it to them. 1 Week later two brand new DVD's including Service pack updates were couriered to me with instructions on use. Even a new product key, although I was able to continue using my own legitimate product key.
Microsoft to my knowledge have not changed that policy. Give them a call, they may be able to courier some new Windows 7 DVD's to you, or maybe even Windows 10 ones; it would be handy though to have the Windows 7 ones.
When I used the DVD's they sent me and reinstalled the operating system I had to download a lot of updates, but other than that all went well and Windows Vista never gave me troubles again.
Regards,
Regedit32