Barb,
Thank you for this summary, this is indeed what it all comes down to.
The only thing I'm not completely sure off is if it happened after a Microsoft update.
I did not check my installed updates and can't remember the exact date the problem first manifested.
The most important thing is that the cover works after a drain, untill restart or closing the cover.
@Ruz: Like I already said, I tried soft resets and hard resets. I installed the newest firmware, I installed both the Windows 8.1 and Windows 10 image from the Microsoft Surface site (which also includes all drivers).
It is not a software issue, I tried all your suggestions already.