Of course, the proper answers are 1) vanilla and 2) yes.
The alternate view would propose 1) chocolate and 2) that any things other than New York style cheesecake or chocolate fudge are neither cheesecake nor fudge.
Now, there is a theory that claims the unmarked case of anything is the common or default instance. Unfortunately this doesn't really help my case - "vanilla ice cream" is a marked case, as are "chocolate fudge" and "new york cheesecake". On the other hand, if someone offers me a piece of fudge without any adjectives, I'm probably going to assume it is chocolate flavored. Ice cream I'd be inclined to inquire further after the flavor, and cheesecake - well, I only learned that what I think of as 'plain' cheesecake wanders around under the moniker "New York" cheesecake a few years ago.
Mind you, recreational fabrication is something that never, ever takes place here...