Yes, it's correct.

Yes, It's. Remember, they're not the same. One is past simple and another one is past participe. Well, I think so.

Yes, it's perfectly natural. We actually say "I'd had..." It's "to have" in past perfect, so the first "had" is an auxiliary verb (which also means you can contract it and add negatives).

Yes, it is correct

It is correct. But most people try to avoid using "had had" as much as possible, as it does not read very well.