Because "prüfen" is a weak, i.e. regular verb. The past participle of regular verbs ends in -(e)t (geprüft, gemacht, gekocht ...). The past participle of so-called strong, i.e. irregular verbs ends in -en and additionally often includes a vowel change (gelesen, gesungen, getrunken ...).
PS: Both German and English have weak and strong verbs. The man who coined the terms "weak and strong verbs" was Jacob Grimm, BTW, who is probably better known for the collection of fairy tales he published with his brother ("Little Red Riding Hood", "Hänsel and Gretel", etc.). Both in English and in German, foreign language learners have to memorize the forms of the strong verbs, whereas the forms of the weak verbs can be deduced because they follow a regular pattern. There is only a limited number of strong verbs in both languages, but because they are very old, they often refer to basic activities such as eating or drinking and are frequently used in everyday life.
1) Weak (= regular) verbs form the past tense and the past participle with -(e)d in English and with -(e)t in German. E.g.
ich prüfe - ich prüfTe - ich habe GEprüfT
I check - I checkED - I have checkED
2) The main characteristic of strong (= "irregular") verbs is that they all have a vowel shift in the past tense. Both in German and in English, strong verbs often additionally have a vowel shift in the past participle, and in German, strong verbs form the past participle with -en. Some strong verbs in German also have a vowel shift in the 2nd and 3rd person singular present tense (e.g. ich lese, du LIEST, er/sie/es LIEST, wir lesen, ihr lest, sie/Sie lesen). Some examples of strong verbs:
ich singe - ich sAng - ich habe GEsUngEN
I sing - I sAng - I have sUng
(NB: The vowel changes aren't always identical in German and in English)
Originally, English also used the "ge-" prefix for the past participle and had special verb endings, but it lost them in the course of its development from a fully inflected language that was mutually intelligible with northern German dialects to what it is today. Interestingly, the -(e)n ending in the past participle of strong verbs seems to have survived in some cases, though, e.g. "I steal - I stole - I have stolEN", "I drive - I drove - I have drivEN".
It's at least much more common to use "Wir alle haben..." when "all" is modifying "we", based on translations of "We have all seen..." on Linguee.
I can't answer whether it's impossible for "alle" after the verb to modify the subject "wir", hopefully a native speaker can clarify that, but in the sentence "Wir haben alle Zimmer geprüft", there's nowhere else that "alle" can be put where it would apply to the rooms.
So I think that means this sentence is unambiguous in meaning "all rooms" -- there's another, more usual, place you can put the "alle" if you mean it to modify "wir", and no where else that you can put it if you want to modify "Zimmer", so you wouldn't write this sentence that way if you meant to modify "wir".
You are right.
To answer you implied question: It is not completely impossible for "alle" to modify the subject "wir", although it is placed after the verb. It depends on the sentence, if there is anything else which could be modified by "alle". E.g.: "Wir beten alle." / "Wir alle beten." ("We pray all." / "We all pray.")
By extending the sentence the focus of "alle" may change. "Wir beten alle zehn Jahre." ("We pray every ten years."): Here "alle" modifies the 'ten years' and not 'we'.
But be careful. "Wir beten alle zu Gott." and "Wir alle beten zu Gott." ("We all pray to God.") don't change the focus of "alle".
The word-for-word translation of "All of us pray." would be "Alle von uns beten." which is actually a correct German sentence, but is not used very often. I could imagine that it is the answer to "Du betest auch?" ("du" is the focus; "You pray too?"), because it emphasizes that each member of the group prays and you consider yourself part of that group.
Correct, it is active and like you said it: "because the subject is I" (and "I" is the person who let somebody do something.)
Ich hatte alle Zimmer von den zuständigen Ämtern/Aufsichtspersonen/Kontrolleuren prüfen lassen. (~the authories checked the rooms while "I" had nothing to do.)
Ich hatte alle Zimmer durch die zuständigen Ämter/Aufsichtspersonen/Kontrolleure prüfen lassen.
proper Passiv: All rooms were checked by the proper authories. = Alle Räume wurden von den zuständigen Behörden/Ämtern/Aufsichtspersonen/Kontrolleuren geprüft. or Alle Räume wurden durch die zuständigen Behörden/Ämter/Aufsichtspersonen/Kontrolleure geprüft.
I change the word "Authoritäten" because it does not fit in our days anymore. Today "Authorität" is more or less only the (powerful, strict) appeariance of a person.
You'll need to provide an example.
One can say either "we checked all rooms" or "we checked all the rooms." Both are grammatically correct, with only a subtle difference. I don't know if the former is accepted by die Eule, but the latter definitely is--it is also the one I'd be more inclined to use. The "the" there is not "extra". Placement of any other "the"s would be extra, and should be marked wrong.