grammatical case - "He had me do this" vs "He had me doing this" vs "He had my doing this"

I know this example sounds awkward, but it’s obviously grammatically incorrect to say "me being here" in sentences like this one:

1. He said me being here was wonderful.

That instance of me being should be my being because we need to use being as a gerund phrase so that it can be the subject of the verb was and so conform to the grammatical structure of Subject–Verb≠Complement.

But what about in this second example?

2. If he had my doing (of) this from the beginning, we would have succeeded.

Because if you say it this way:

3. If he had me do this from the beginning, we would have succeeded.

Now there seems to be a conflict of had and do because both are verbs. Can someone explain, please?

Here is the NY Times grammar example: the After Deadline blog posting of 2012-05-29:

Aurélie Filipetti, 38, a novelist and a legislator, was named culture minister. She had made public an account of being groped by Dominique Strauss-Kahn, the man whom many thought would be the Socialist candidate.

They say it should be who and not whom because the example here is that it's ultimately the verb being done by the subject that counts, not the "many thought" part (seems the verb gives priority to the subject rather than the other way around).

Answer

Those are three different problems.

1. "He had me do this” vs “He had my doing this”

The former is correct. The latter is nonsensical.

2. 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.

3. 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.