First of all, what happens to his shield? Does he keep the +1 armor bonus in bear shape? The "you and your possessions meld into a perfect copy of the species’ form" doesn't address whether it means that your equipment effectively disappears until you revert, or not. Only that they "meld together".
The vagueness is deliberate.
Maybe in your Dungeon World the druid's possessions do meld and reform, so you get a bear with metallic plates (the reformed shield) on it's flanks and haunches. Then, yeah, it still has armor.
Maybe in your Dungeon World it really does completely become part of the new form, in which case it's not worth armor anymore.
Neither will break the game, both have fictional implications—if your shield is still visibly defending you in some form, you're not going to be able to blend is as just your average bear, are you? In general, follow the fiction: if the shield is still in some way protecting the druid, it's still worth armor.
Second, in case his shield magically and momentarily dissappear, would it then be a move that requires expenditure of hold to "Hack and Slash" as a bear, since you make use of it's animal features? If the shield disappears, his weapon would too, after all.
Yes, his weapon melds in the same way—maybe disappearing completely, or maybe becoming in some way embedded in his claws. No matter what his range will change.
The thing is, the druid is now armed with bear claws. I'm not an ursine expert, but I'd say they're at least as much a real weapon as a dagger is. The druid is still able to hack and slash as long as they can trigger the move.
Third, how do I as a GM decide the associated moves? Just on the spot? I would have preferred that the rule looked more to the player to explain what he wanted to achieve. I had a hard time coming up with a move for his "Bear form", since he didn't have a reason to use it, except for the coolness factor, but he kept wanting to know what he could do with the form.
As a GM you ask questions, and this is a great time to do it, which it sounds like you did. When they take on a new form ask "so why are you taking on this form?" It'll give you an idea of what they think a sparrow can do, so you can give them moves or say "I don't think sparrows can lift halflings" or whatever.
In this case, it sounds like the player basically replied with "because it's cool" which is a weak answer, but still an answer. The thing I really pick up on there is that bears are cool—intimidating, awesome, impressive. So I'd give them a move about that: "Terrible growl" or something.
Then there's the obvious thing for a form taken in combat to do: "maul them" or "rip something apart" or whatever.
Then I might tack on one more to twist it a bit: "Find a source of food" perhaps. Or, like Antisinecurist said, "crash through." This is a potential to give the player the unexpected, but it's not really required.
So I guess that's a bit of a formula: what the player wants, the most obvious thing, and a twist. It's not perfect, but it might be useful to get started.
A note on shapeshifting in general: taking on a new form is, in a way, saving up successes for particular tasks. The druid makes one roll and, through the clever selection of a form, can turn that into 1 or more successes. The cost of this is the new form: since everything is triggered by the fiction taking on a new form changes what rules engage. That's the fundamental tradeoff of a new form: that form's strengths turn your one roll into more successes, but that form also limits your options.