The real question is how will SBUX react to this... if they start rattling legal sabers, it could be bad. However, if they go along with the joke, then it could be great publicity. I hope they go along with the joke.
Given that this is Comedy Central (owned by Viacom), I'd imagine the idea was run past Starbucks' legal department before it was greenlit. I'd also wager there's a legal agreement governing what CC can and can't do, and even how Starbucks will or won't react.
Oh, I doubt it. This is probably produced by an independent production team for CC. I don't know if CC legal would be notified before the fact of this or just prior to broadcast.
In the OP, they had quoted Starbucks (PR?) saying that they thought "Dumb Starbucks" was misusing their trademarks.
Viacom is all about the money - I'd say there was a non-zero chance that they actually made Starbucks pay to be the subject of the sketch, and Starbucks PR are just playing along.