Help with Possessive + Gerund to express the French Subjunctive?

Hello, I'm an English learner. Need your help, please. Could you tell me when I should use "me" instead of "my" to express the French subjunctive? Here are some examples to make it clear: 1- "MY being" here makes him feel better. OK 2- You can count on "MY doing"... show more Hello, I'm an English learner. Need your help, please.

Could you tell me when I should use "me" instead of "my" to express the French subjunctive?
Here are some examples to make it clear:
1- "MY being" here makes him feel better. OK
2- You can count on "MY doing" my best! OK

Now my problem is:
3- He came in without "ME noticing" => Could I also say "without MY noticing"?
If yes, in which case do we use "me" instead of "my"? (same hold for the other pronouns)

Thanks a lot.
Update: To ihrtshchr: Thanks for answering. :) So, in all cases dealing with this structrure, can we say that "me" is informal" and "my" a bit more formal? Like in these examples: - Do you mind their / them smoking? - I’m not annoyed with your / you being here. Is it correct to use both... show more To ihrtshchr: Thanks for answering. :)
So, in all cases dealing with this structrure, can we say that "me" is informal" and "my" a bit more formal?
Like in these examples:
- Do you mind their / them smoking?
- I’m not annoyed with your / you being here.
Is it correct to use both like that?
2 answers 2