Professors McGonagall and Moody kept them working until the very last second of their classes too, and Snape, of course, would no sooner let them play games in class than adopt Harry.
I think "no sooner let them play games in class than adopt Harry" is a metaphor here. It's implying that Snape would never let them play games in class. It's because I understand Snape hates Harry very much from the book, so we can't really expect him to adopt Harry(I think 'adopt' here means "take one into your own family and make one legally your son or daughter"). In my opinion, it can be paraphrased as "Snape would never let them play games in class just as he would never adopt Harry". Is this understand correct?