Those are three different problems.
- "He had me do this” vs “He had my doing this”
The former is correct. The latter is nonsensical.
- He said me being here was wonderful.
Yes, logically, it should be my, but no one ever says it like that. It's always me in this case. Go figure.
- She had made public an account of being groped by Dominique Strauss-Kahn, the man whom many thought would be the Socialist candidate.
You're quite correct here. It should be who, since this is the nominative case. Some people who don't bother much about cases, though, reckon that "whom" sounds more sophisticated. That's the only explanation I can think of, and should anyone have a better one, I defy them to prove it.
ADDENDUM (UPON POPULAR DEMAND):
FumbleFingers' example, but the teacher obviously had my cheating in mind has nothing at all to do with any of this. My terminology is sketchy, but ...
My teacher wouldn't have me cheating at the exam means he'd kick my ass if he caught me in the act of cheating.
My teacher had a problem with my cheating means he knew I was a cheat in general and had a problem with it.
He had me do this means his purpose (which he achieved) was to coax, force, or trick me (not force my, which would be nonsensical) into doing "this."
MORE ON THIS (from the OP's puzzled comment):
If the teacher had my cheating on the exam, she'd fail me. This works, right?
No, it doesn't.
One: To have someone [the accusative case] do something means to encourage, require, order, or force them to do it. It wouldn't make sense for the teacher to ask you to cheat and then fail you. I mean, shit happens; but generally teachers fail students because said students do something the teacher didn't ask them to do.
Two: "She had my cheating" doesn't make any sense unless it's the title of a book. As in "The teacher had My Teaching, by Umberto Eco, in her hands."
If he had my doing the work: This is wrong, and (attention! important information!) it is NOT related in any way to phrases like "Me being there brightened up the morons' lives." Nothing at all. Different rules, completely.
You can have me, or Linda, or John, do all the work for you.
You cannot, I emphasize, cannot, have my, Linda's, or John's, do all the work for you. You can't have Linda's do all the work for you. You can only have Linda do all the work for you.
Please tell me if you still have a problem understanding this. We'll post a whole new question together and have another spirited crack at the son of a bitch.