Hello Lenka,
To me, both #3 and #4 are correct.
The structure in #3 is possessive + gerund ("my X-ing"); the structure in #4 is object pronoun + participle, e.g.
5. I hope you don't mind
my asking
— "my" = possessive, "asking" = gerund; "my" qualifies "asking".
6. I hope you don't mind
me asking
— "me" = object pronoun, "asking" = participle; "asking" qualifies "me".
However, #6 is a disputed usage: there are some teachers, authors of style guides, and grammarians who believe that it's incorrect.
Here's
one such person.
As I say, I don't myself believe that it's incorrect; and it's a common structure, in ordinary spoken English. But if you're taking an exam, or applying for a job, it's as well to know that there
are some people who dislike the structure!
All the best,
MrP