Object pronouns + verb+ing

Question

As non-native speaker of English, I'm having trouble making sense of a structure pertaining to object pronouns:

1. Likelihood of me doing this ...
2. Your plan involves me attempting to prepare plans necessary for you.
3. Chances of me getting into that school is high.

What is it that establishes the connection between object pronouns and verbs in gerund form in terms of both grammar and meaning?

In my language, expressions corresponding to the expressions in bold above are constituted with "my" instead of "me". That is the reason why I'm having trouble.

Answer 1

That's not the gerund.  That's its identical cousin, the participle.  If it were a gerund, then you would use the genitive form "my":

• The likelihood of my doing this is small

In this example, "doing this" is a gerund phrase.  "My" modifies the gerund phrase.  The modified gerund phrase is the object of the preposition "of".  
 

• The likelihood of me doing this is small

In this example, "doing this" is a participial phrase.  Participial phrases can modify nouns.  This participial phrase modifies "me".  The modified objective personal pronoun is the object of the preposition "of".  
 

Because there's no visible difference between the gerund and the so-called present participle, it almost looks like "me" and "my" are doing the same job.  They aren't.  The "me" acts as a modified object; the "my" acts as an object modifier.  Of course, in both cases you end up with some modified object for the preposition "for".  The overall meaning of each is much the same, and the choice between them is often no more than a question of stylistic convention.

If it helps, my personal preference is for the gerund construction.  I prefer the action, rather than the person, to be the literal object.