As I don't have a Touch or Type cover yet I was keen to try and switch off the display, if I knew I was only going to be putting it down for a few moments but wanted to try and save on battery life. So, I tried changing the Power button to "Do Nothing" hoping it would still at least turn off the display. It didn't work. The button now "does nothing".
Unless you require processes to run in the background, your situation is best served by setting the power button to sleep and "don't require a password" for faster access. The wake-up time from sleep is instant anyway.
I have mine set to "do nothing" when I close the lid, and the screen stays on when I close the type cover. Are you sure the screen is turning off on yours? I can peek in the gap and see the screen is still on. Maybe the backlight on yours just autoadjusts to a very low level? Do you have any other settings changed? Curious as to why our two are working differently.
Also keep in mind that the Surfaces all wake from sleep in less time than it takes to fully open the screen on most laptops, so I don't see why this is a big detriment.
My screen definitely turns off with the cover closed. (CP > Hardware and Sound > Power options > "Choose what closing the lid does" > Do nothing) You did remember to select "save changes," right?
Otherwise, if ours are set up the exact same way, I have no idea why it would be different; mine is a Surface 2 with Type 2 cover.
I believe the OP wants the Surface to remain "awake" for background processes to run; that's certainly the only time when you'd want only the screen to turn off.
@OP: About the multiple screen scaling, I have no idea, though. There are other threads about that.