Quinn comes out of Burke's office pissed off because Burke told her that it is over. He's getting divorce and probably is out of there, not to mention his wealth will be halved. She didn't expect it to end so suddenly, probably having her own long time plans which she has to abandon now. That is why she told MC at the BBQ (don't remember exactly) that she is not there for studying and will probably be gone next year (she already knows about Burke quitting).
I didn't think of that but you are probably right. It really makes sense in regard to the conversation between her and the MC about quitting college. Even if attending college was always just a facade for her, main activity, she nevertheless really wanted to become the HOTs' president next year after Sage. It's very apparent in all previous episodes. But without Burke's protection, she probably wouldn't be safe to continue the 'restaurant' business, thus losing a major source of income. Continuing college wouldn't be feasible for her anymore.
But I don't think the reason for Burke's withdrawal from the 'restaurant' business is his divorce. There is no reason for him to lose his position in college's faculty because of a highly personal matter such as a divorce. Let's remember that Burke called Quinn ('Priscilla') last episode and arranged a meeting with her just after his conversation with Maya. He heard from Maya about the 'free tuition at the HOTs' rumour and probably thought that the prostitution business got out of hand if even his daughter's friends are talking about these rumours. And with the Mona situation, it simply became too dangerous for him. He doesn't want to take risk anymore and that's why he arranged to meet Quinn/Priscilla to tell her about his decision.
As for other points mentioned in your post, I agree with them 100%.
Yeah, that's right.
What if whoever sent him Jade's MC video wanted that reaction?
So he could divorce Jade and leave college?
If that's the case, it's clearly not Vinnie or Quinn. They are clearly interested in Burke as a patron of the restaurant and drug sales.
One of the DIKs wanted him gone.
Burke is only a patron for Quinn. Vinnie doesn't need him because he doesn't operate in college. I'm not even sure Burke knows about the drug business. It's possible he only knows (and protects) Quinn's prostitution business.
If Vinnie got that video from the DIK laptop (by himself or with someone's help), he could send it to Burke which is in my opinion the most probable scenario. His goal was to sabotage the DIKs for whom he has a grudge since he left.
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