Is it correct to say had had?

Answer 1

It depends. If you use it in a sentence like "She had had the food taken from her so she was starving" or something it's okay.

Yes.

Past perfect is had + past participle and sometimes had - which is the the past participle of have - is used.

We had had a nice day until James stood on the wasp nest.

The army had won the battle before they crossed the border.

Past perfect is usually used to talk about something that happened in the past (past perfect) before something else that happened in the past (past simple).

In the first sentence had a nice day happened before James stood on the wasp nest.

To more examples to make it clearer.
We had taken tea before the sun rose up.
We had had tea before the sun rose up.

Answer 2

Yes, "had had" is correct when indicating that someone possessed something in the past. For example, "He had had a car before he moved to the city."