Yeah, I realised as soon as I wrote the question that the Internet would be full of rage if the opposite were true. It's just that, sometimes in the past, there's been "Are you serious?" moments with products lacking what I'd consider basic functionality. iPhone comes to mind.
Yes you can. I play with my DS4 on PC, actually, with both charging and signalling over USB (don't know if wireless is possible with PC, but I don't care, really. DS3 did it too btw. As you said, would be unacceptable otherwise.
Can you just use the DS4 plugged in all the time? Of course you can, I can't believe I just asked that.