1°) It is to be noted that the English perfect and the French perfect have developed pretty much in the same way and at the same time for reasons that are known : the influence of French grammar and syntax on old English, like 'of-genitives' (instead of saxon -es/'s for example). However the French perfect has become a real past tense in French, now called "le passé composé".
You can understand the parallel between "I have it done" (passive) and "I have done it" from intuition. But there is interference with modern grammar in which "I have it done" means "somebody (else) does it for me." Yet, the original model of the syntax meant (roughly) : "I have it done (by myself)", that is is located relative to 'I'. The repetition of the subject pronoun makes it disappear in this structure because it is not a form of insistence. Therefore only the object remains to be spoken about and we obtain what must be called "a passive participle" do + -en = done.
In the "active", 'do' (for ex.) means that you find the agent to the left (or before do in the spoken chain and they object to the right (or after the verb).
The passive flexion -en simply means that the order is reverted:
done means that you find the object to the left (or before do) and the agent to the right (or after done) in the spoken chain.
It indicates the reversal of positions.
Now the perfect is a complex structure with two verbs.
In the logical order of operations, the first is the lexical verb, here do, which is affected by -en. The consequence is that the object becomes the topic and the agent can even be deleted.
The second verb is the operator have, which in its formal use is not lexical and expresses location relative to the subject of the sentence.
Contrary to the active do, the passive participle done means that the process of the verb is finished (probably because the topicalised object is now fully achieved and clearly defined.
This is how the perfect aspect (not tense) is still present and expressed the benefit for the subject of a completed process (typically) in the form of its completed object.
Now, the syntax of English (and French) has always easily managed auxiliaries (be in the passive; modals; now do/does/did, etc.)
So what happened when the combination of this simple passive form was combined with have was not that the object took up the final position :
I <have done> IT.
But rather, it was that the passive participle combined with what became an overt auxiliary have in the Verb Phrase:
I have <DONE IT>.
For many verbs that were not transitive (no object) or reflexive, in early Middle English, the auxiliary was be, as in French the auxiliary is être ('be') for many verbs of this type : “Je me suis lavé” = I have washed, etc.
This still exists in English in very rare expressions such as: "I'm done" meaning 'I have finished' (not 'I'm nearly dead, of course!) or: 'I'm finished' = 'I have finished' in standard English.