1 Apparently it's been done. (The tl;dr is that they're not bad, if you like crispy as opposed to puffy cookies.)

A different experimenter tried other kinds of cookie dough ice cream, i.e. lesser flavors that the One True Cookie Dough Ice Cream, which is of course Ben & Jerry's. I give this person bonus points for re-freezing the melted ice cream and then eating an ice cream sandwich using the cookies from the original ice cream. Yum!

2 I heard of odder things - US food processing companies have been known to buy frozen pastries from Europe, ship them to the US, melt them down to extract the sugar which they will use in their own food items, and toss the rest.  Because the prices of real sugar made the process cost-effective.

3 I suppose it depends on the whim of the writer, and what sort of cartoon this cooking exercise is in. It is unlikely to last more than a single episode unless, perhaps, the protagonists are doing a comparative study of different brands and then it is likely to be a background plot. Of course there is the post apocalyptic scenario where the heroes discover a quantity of cookie dough ice cream lodged in a glacier and due to a preexisting quest to obtain chocolate ship cookies decide to do this. With what I'm seeing from this new band show, I'm not going to bet against Kyo-Ani being able to make a cartoon like this work. 
Wait...What was the question again?