The sentence "We have had the house painted" contains the causative verb "HAVE" in its present perfect form. A causative verb is used when the subject (one person) "activates" another (second person) to do something. The subject does not perform the action himself but rather "causes" another person to do it for him. As in your sample sentence, the subject "WE" did not do the painting themselves but they asked somebody else to do the painting for them.
Thus, "have had" in your sentence can be labeled as follows:
HAVE (Aux Verb - for present perfect) + HAD (Main Verb - past form of "have")
If we were to conjugate the verb in the different tenses, it would appear as follows:
PRESENT: We HAVE the house painted every year.
PAST: We HAD the house painted last year.
FUTURE: We WILL HAVE the house painted next year.
PRESENT PROG: We ARE HAVING the house painted right now.
PAST PROG: We WERE HAVING the house painted when it rained.
FUTURE PROG: We WILL BE HAVING the house painted next week.
PRESENT PERFECT: We HAVE HAD the house painted red and green once.
PAST PERFECT: We HAD HAD the house painted red and green already before she decided to choose blue and yellow.
FUTURE PERFECT: We WILL HAVE HAD the house painted by the time he gets back from his trip to Europe.
As to the word "painted," yes it is a verb in the past participle form, not functioning as an adjective, but as a verb in the passive voice. If we were to identify the DOER in your sentence as follows, it would clearly show that "painted" is not descriptive.
We have had the house painted BY THE CONTRACTOR.
I hope the above explanation helps. I recommend Berlitz' English Verb Handbook as a reference for the nitty-gritties of English verbs. :)