if your case doesn't have a built in screen protector then I would say yes it is necessary.it would definitely be the smart thing to do as it is low cost and easy to install, albeit a pita. a good screen protector should not take away from the experience.
I did try several but didn't quite like them enough to keep them on. I just really like the feel of the raw glass on my fingers. (BUT I'm extremely careful with my tablets and since loosing my job, I'm not out and about much) I do advise others that they're a good idea for active people....accidents happen. In other words, you might have to buy a few variations till you find the one that's right for you, but it will be money well spent. It would be heartbreaking to get a nasty scratch on such a fine looking screen.
Do you mean the Daydream function? You're sure it's switched on? Go to Settings>Display>Daydream, and make sure the blue switch at the top of the screen is set to On.
Your success is going to be mixed. Moving apps to the SD card is no longer supported by the OS, so you would most likely have to root your tablet. Plus, the app developer in question would have to support the feature as well.