Do dragons still pick people up and drop them like in the video when the dragon picks up the giant. I have been playing for awhile and haven't seen it happen and I'm wondering if it is still in.

Yes, I don't know if the one i saw was a scripted event, but it was during a prominent point during the MQ.

Yes. It's happened a couple of times to me. (to be fair though, my first few dragon fights I tried to pull to inhabited areas to help kill it.)