As a ?Maiar? who has been traveling Middle-Earth for centuries, steeped in the languages and lore of every race and individual he came across, it is granted that Gandalf would have many aquzintances and connections. Middle-Earth being a land of numerous beings, and gandalf's travels being as broad as they were, it is more than likely that, at some point, he would have encountered one of these giants. After learning their language, it is quite possible that Gandalf could have developed a friendly relationship with one or more of these giants. It is also possible that a giant may be indebted to Gandalf, who, being a wizard and all, may have used his wisdom or powers to aid a giant in the past.
Gandalf, therefore, would not have to "command" mountain giants. He would only have to make the request that the giant plug up the whole so that travelers are not enslaved or eaten by the more uncouth inhabitants of the subterranean. Any decent giant would be more than willing to render such a simple, yet nonetheless beneficial, service to the greater population.
Although it is also possible that, during his many escapades, Gandalf may have performed some deed that would have earned him leadership over the mountain giants. This is unlikely, however, because Gandalf seems to have preferred to remain politically neutral so as to prevent the complication of his already complex and numerous duties as a semi-divine guardian of what is just and good.

Me too.

And not having any of Saruman's blasting powder, or whatever it was that Saruman's Uruks used to open the hole in the outer wall at Helm's Deep, a giant was probably the best individual to collapse the tunnel openings. Dain's Dwarven Warriors assuredly would also have been able to do the trick.

But I have an idea that Gandalf's line was a 'throw away' line and he never followed through with the project, due to a previous, and more pressing engagement at Dol Guldur.

Still, something must have been done, for the Beornings were able to keep the Mountain Pass open circa Bilbo's Long Expected Party.

Radagast, like Beorn, lived in peace and friendship with the other creatures of the world. That was a common bond between them that was conducive toward friendship. When Gandalf mentioned Radagast, Beorn realized that he was speaking with a friend-of-a-friend, which is slightly better than a stranger.
As for Beorn's shape shifting ability, I think Tolkien meant to leave that to the readers' imaginations. It would make sense to me that, early in Middle-Earth's history, a man may have somehow found favor in the eyes of a Valor, who may have given him the ability to help him out of a tight spot. Being a bear can be very convenient at times, such as when the weather is dangerously cold or you are unarmed and surrounded by unfriendly beings. That man who was given the ability most likely passed it on to his children. As far as I know Tolkien does not give an explanation, the mystery only makes the story that much richer and more unpredictable for me.
You should also keep in mind that Tolkien wrote The Hobbit just as a story. Years later, he decided that he would use it as a legitimate chronicle of Middle-Earth. Because it was originally intended to stand on its own, the book may have a few loose ends that Tolkien never got around to tying off.