Harvard "liberal"??

My impression of Harvard is that it is wildly conservative, at least their business and law schools.

Not the law school. The business school, sure. But Harvard College (the undergraduate school), and many of the graduate schools in the humanities, are regarded as intensely liberal by virtually all of the GOP. And that assessment is probably more correct than wrong - as a general matter, the faculty and students at Harvard are overall significantly more liberal than the country as a whole.

And...again...why does he care? He's a lame duck. He can spend all his time enriching himself, and nobody can do anything about it (thank you, Justice Roberts).

I can't answer to his psychology, except to point out that he obviously relishes aggressive conflict with people he perceives have wronged him, or even merely challenged him. It's his MO - if anyone takes a swing or a jab at him, he fights back as hard as he can.

But it's obviously in his best interests to have these fights. He wants to keep control of the GOP, so that he can do what he wants. He doesn't want anyone in the GOP delegation in Congress getting any ideas about pushing back on him. And he absolutely is focused on the midterms and trying to make sure the GOP keeps control of the House, so that the Democrats don't have any ability to start investigating all his methods for enriching himself.

Why would he care about Democratic investigations, when he can't be put in jail? Because he can't enrich himself on his own. He needs to have people giving him money, giving him valuable opportunities, giving him planes. Democratic investigations can't really affect Trump, but they can make life much more unpleasant for his counterparties - which makes it harder for him to do the type of deals he wants. It's much easier for him to have his little crypto shindig (for example) when his guests know that there's no chance that they'll get hauled in front of House subcommittee to testify under oath any time soon.