New motherboard

If this is a retail version of windows, you can clean install using your install key. If this is an oem version ie what comes on a dell, hp, whatever, then you need to purchase a new windows lic. OEM type lic live and die with the original hardware. You change the motherboard and you need a new lic or a new copy of windows.

 

With your original motherboard NOT with a different board. You do not need a key to install win10. You need a key to activate win10 for the first time. Each install after that will auto activate as long as you do not change hardware.
If you are upgrading a dell, hp, etc; you need a new key.
Is this a big box case you are attempting to use ie was this a hp, dell, etc? If so, you may need a new case as well. Dell, hp, etc stuff is proprietary that is it is made to the mfg's standards NOT industry standards. A standard atx motherboard may or may not fit a dell case; pci slots may or may not line up, pw supply may or may not work.

 

To answer the original question, there is a guide at the top of the forum for swapping boards without a clean install. If you follow all of the steps of the guide, you will not need to clean install. You will however have to activate and an oem key is not going to activate with a different motherboard. So yes, you can reuse your operating system however you still need to purchase an install key. Most of the time most people would just clean install however that is up to you.

 

Probably newegg. There are a LOT of so called cheap install keys advertised for sale. Most of the time these do not work. Just purchase one from a reputable seller. I just checked the egg and they have win10 pro for $149 and home for $109.
https://www.newegg.com/Operating-Systems/SubCategory/ID-368