Yes, your rewriting is correct; the original form is grammatically incorrect. In that first sentence the first use of the verb 'to have' [ 'he has not *had . .' ] is as an auxiliary to determine tense ie present perfect. * 'have' should be replaced by 'had', the past participle. The second use - 'have' [which should be 'had'] indicates that the job will be done by someone else. To 'have something done' means to arrange for it to be done. In the correctly rewritten version the tense [present / future] requires the auxiliary verb 'to do', and the subsequent use of the same verb ['have it done'] is coincidental. You would still use the first 'does' with any other verb in the second phrase eg “If he does not have it fixed, he will fix it”.