Okay update:

Started the demo download at 2am and it finished at 2:45am. And at 4:19am it completed downloading the update for the demo. So it worked.

Why in the hell do all of my disc games not seem to do this?

Cook food, return when done. ;P

It seems to be working for me. So you're not seeing anything in your notifications that says [insert game name] has completed installation when you boot up from standby mode?

No for my games I always see something like "Battlefield 4 was added to downloads"

Yeah, I've had that too.

My standby doesn't work as advertised either. Systems never updates, games update sometimes, and purchases from the PSN store don't download onto the system.

Sony needs to get on this and start doing as advertised.

Since people are answering questions, maybe somebody can help me with my PS4 problem, too. I played Tomb Raider on the Vita with remote play but the back touch screen didn't work. It registered that I was touching 'em because a little pic of a Vita would appear showing what I was doing (if that makes any sense) but the "buttons" didn't do what they were supposed to.

Little help, please?

It's because you have it in a horizontal position. It's called a STANDby mode for a reason.

Yeah, I was touching it in the right places.  Like I said (poorly worded), when I touch the rear, a little pic of a Vita would appear on the screen showing that I was touching it in the right place but nothing else would happen.  I didn't see an option to check the configuration on the PS4 aside from the type of connection I was using.  On the Vita, perhaps?

And let me say how awesome it was seeing such sharp graphics coming from a handheld!  It almost moved me to tears!!

I don't think the back touch pad does anything with Tomb Raider. Hold the front touch screen to view the map. I believe you just tap the touch screen to go into your instinct mode. To find out the button config for any remote play game, hit the Playstation button and tap the button guide.