Is there a place I can leave my suitcase while I shop? I'm out of my hotel with hours to spare before the airport. I want to shop while I wait.

You can usually ask your hotel to hold your luggage

Our hotel held our stuff until the evening.

There is an information desk which near Earl of Sandwhich. They have helped with that before. But it does depend on how big your suitcase is

Yes

Check guest services at the town center

I believe there are lockers, in the middle. But you can double check with a Disney cast member!