Dreaming about a game is really perfectly normal. Dreams tend to take the events and things you think about over the past few days and use them for themes/visuals/etc. Anyone who plays or even just thinks about a game is viable to dream about it and it's not just the case of "omg there must be nothing else happening in your life to dream about a game" or "omg, you must've played 17 hours a day for weeks to dream about games". So I call bull on that, but yeah, if you wanna count it, I've dreamed about games and I've dreamt dreams that were about something else using game mechanics, visuals, themes etc.
On the actual topic... yeah had that plenty of times. For example, Assassin's Creed had the effect of me shoulder-pushing-aside people when in crowds. Still kinda do it, it's surprisingly effective
On the music humming, oh God, that was the worst. Jedi Knight: Jedi Academy. I remember it cause I played it loads, both home and at a cybercafe with some mates in multiplayer. The thing is, the tune of the multiplayer got so stuck in my head, that when I was in a cybercafe on one occasion, I was humming it, barely noticing myself cause I had the headphones on. The guy next to me playing WC3 turned to me after some half an hour or so looking absolutely desperate and basically completely politely said "could you please shut the fuck up?"... I suddenly realised what I had been doing and was embarrassed as hell ^^