The dog is welcome in the surrounding park but not in the museum.
No
The good thing about this space is, there's lots of it indoors and out. Locals bring their dogs and kids and hangout at the walking trail and park behind the museum. You can partake in owl and porcupine visits when they come outside and have a picnic while you are visiting!
I don't think so. It might scare all the birds they have there. But you can always call and ask them. The staff is super nice.
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