Is there any possibility to completely disable the new Edge browser?
The old IE could be disabled in Settings → Default Programs, but in Windows 10, the checkbox is missing for Edge.
Is there any way to get rid of Edge, or at least to disable all entry-points?
Best Answer
UPDATE AUGUST 2018: Microsoft keeps changing things a bit, but you can still "Disable" the Edge browser by doing the following:
Edge should now be "disabled".
Keep in mind that Microsoft may try to re-enable Edge during a future update...
THE ANSWERS BELOW ARE OLDER ANSWERS...
UPDATE: In January 2017 I disabled Edge on a new Windows 10 install by simply renaming the "C:\Windows\SystemApps\Microsoft.MicrosoftEdge_8wekyb3d8bbwe" folder. (If you are asked to allow administrator privileges be sure to answer in the affirmative.) Renaming this folder was not possible in the past without following the more detailed steps below. This much simpler approach achieves the exact same results as the more complex instructions below! However, for older installations and/or Windows upgrades the below instructions may still be the only answer.
THE ANSWER BELOW THIS LINE IS THE OLDER ANSWER...
As already stated, the other answers are no longer valid due to changes in Windows 10 -- Microsoft has removed all supported methods for disabling Edge. As of today, this answer still works. Although parts of this answer were given, not all of it was given together in a step by step manner.
Been running Windows 10 Pro like this on 100+ PCs for several months without issues.
A NOTE REGARDING WINDOWS ANNIVERSARY UPDATE: The Anniversary update appears to undo the above changes and re-enable Edge. In my case I just ran the above steps again, and Edge is once again disabled.