Past Participle

To have + past participle is "the perfect infinitive". It's used to show that an action happened previous to some other action.

I had to have taken an umbrella, for example, is like saying It had to be that I had taken an umbrella. (The only logical conclusion was that I had taken an umbrella -- before I discovered something else, for example, that I needed an umbrella.)

need to have done to it is a different case. It means that somebody needs to do something to it. The 'something' is missing because it has become the interrogative pronoun what. "It needs to have 'what' done to it." Because of the intervening implicit 'what', this is not a perfect infinitive like the rest of your examples.