Is anyone aware if Fallout 4 will have streaming restrictions?

After release, probably not.

Before Skyrim, a lot of people, I don't know how, had copies and were streaming before Beth cut them down.

What about issues with the Music on the in-game radio?

Shouldn't have any problems legally.

Bethesda has always allowed their games to be streamed after they've been released, so I imagine FO4 will be the same.

That would be between you and your streaming service, not Bethesda.
I know Twitch allows any music to be played during your live stream, but a lot of music will get your archived videos permanently muted.

Apparently several people received their copies on the 360 early due to a snafu with the store they'd pre-ordered from.

Lot of threads got locked/deleted that week and a lot of warnings handed out too I imagine.

I managed to sneak into a few streams before they got executed. Some would get as far as say three hours in before it got cut.

As for the music...Well, I may be wrong, and please correct me, some of the music on the radio is very old and may have passed into public domain by now. Granted, the radio would hinder any commentary you might due, dueling audio.