Yes it is. The meal they featured on the show is at the top of the menu and delicious

No private parking. All street parking or pay lot. Usually not a problem finding a spot but don't leave your car over night in woods hole it will be towed. I think meters take card 2 hour limit.

The place changed ownership a few years back. I'd be guessing, but that makes the answer to your question "probably not".

Have no idea there's never been a dog there when I've been there give them a call on the phone they'll tell you