Verified answer

You want to write "insists on".

Next, "me" is fine. Some people, especially teachers of English as a second language, prefer "my" because they think that posting is a noun. But posting is a non-finite verb form (a present participle) and takes an objective case pronoun subject.

"She insists on me posting that picture on Facebook"

I think the first one but just add "on", so it'll be "She insists on me posting that picture on Facebook"

The second one(:

Actually none of them is completely correct.

I really don't get the first one.

If you add "on" to the second one, it'll be correct.(and natural for sure.)