Firefox and Edge use two different Flash versions, and if Edge is in use should not effect the Firefox variant.
I successfully installed Firefox and the NPAPI version of flash on Windows 10 both during technical preview and on my office machine running windows 10 as well without a hitch
My suggestion is to download the offline installer at the bottom of the page here: Installation problems | Flash Player | Windows
A) shut down and restart the machine OR
B) type in cmd in the search bar at the bottom, right click on the selection and choose run as administrator then type the following lines one by one and press enter after
1) taskkill.exe /im firefox* /f /t
2) taskkill.exe /im plugin-container.exe /f /t
3) taskkill.exe /im flash* /f /f
Then try running the installer. Also there are processes like steam browser and battlenet ( game browsers ) that use flash. Skype I hear uses flash ( not sure which plugin type ) and I have seen yahoo messenger use flash player.
Hopefully this will resolve your issue