Is this an official Back to the Future product?

The listing does not say. Treat it as fan merchandise unless the seller states a license. Official Universal-licensed items usually carry branding on the mug or packaging.

Can it go in the dishwasher or microwave?

That depends on the print process. Many printed novelty mugs are hand-wash only to protect the design, so check the seller's care instructions before buying.

Standard ceramic mug size is typical for this kind of merch, but the exact capacity is not given. Ask the seller if capacity matters to you.

Will the print fade or peel?

Sublimation-printed mugs usually hold up well; decals and cheap transfers can fade. Reviews for this specific mug are not available, so judge by the seller's print method and return policy.