Is there a petting area? If so is it open when it rains?

It's outside but there are many areas to stay in shade or when it rains, and there is a roof over the place next to the animals

There is a enclosed pen where you can pet sheep or brush the goats. It is outside so you might get wet if it rains.

Yes

If you're looking to pet wet animals try seeking an aquatic nature center

Yes and yes