1. Despite/ In spite of his being a good man, I don't like him. Either form is correct. The 'his' is also correct. 'Being' is a gerund, so you need the possessive adjective 'his'. Wendy has a point, though, when she says that 'him' sounds better than 'his'. In fact, 'him' is wrong, but it is so widespread among native speakers that it 'sounds' more right to most people than the correct form. 2. Despite (my) having prepared carefully for the test, I still failed it. Can I drop "my" in this case because the subject in the clauses is the same one? Yes, exactly right. You can omit the 'my' here because the subject is the same in both clauses. This is why you can't omit the 'his' in the sentence above. You have understood a grammatical rule which the majority of native English speakers are unaware of. 3. Although I can not be by your side/ be with you on your birthday, I want to spend time celebrating your special day with you on Skype. 'With you' is more natural sounding. 'By your side' is a bit 'cheesy'. We don't normally write 'can not'. We either use the combined form 'cannot' or the contraction 'can't'. And yes, you have a good grasp of English grammar. Better than some native speakers!

"Despite / in spite of" are the same and can be used interchangibly. I don't know which is grammatically correct, but "him being a good man" sounds better to me tha "his being a good man"

"cannot" is much more common than "can not". But both are ok. Your English is very good!

"be by your side/ be with you". Both are fine. "Be with you" is probably more common in the US.

How about the other sentences? Are they correct?