Here is my take on the whole thing, I will try to answer your questions as best as I can, and correct me if I am wrong anywhere:

1) "ArchThordan" had a tremendous amount of prayer. 1000 years worth of prayer. While those in Ishguard did praise Halone and seek her out for strength in times of need, it isn't the same as praying to her. The Holy-See were more than anything followers of Thordan I. The Dragon Song War had nothing to do with Halone. Ishguard were getting behind Thordan and believed enough in him to continue feuling the war for 1000 years thinking the dragons are the ones who drew first blood and more than anything wanted retribution for that.

2) Halone would have been just as strong as Thordan was, with they same amount of prayer and Aether thier power would have been equal.

3) Because it was Thordan who was revered by the Holy-See. It is also my belief that the Archbishop did not wan't to actually become a god, but the hand of a God. Like the Garleans, the Holy-See had the same goals as we do: Extinguishing the Primal threat. The thing is they were fanatics and were going about it all the wrong way seeking supression instead of freedom. He wanted to do Halone's bidding, not become Halone herself.

4) You seem to be confusing worship and belief. They don't exactly need down-on-your-knees-worship in order to fit the conditions of summoning, they simply need a strong belief in place, and as I stated before Ishgardians BELIEVED in Thordan I. This belief is also separate from the Aether supply, both belief and aether are required for summoning. The aether supply was the Eye of Niddhog. If you remember we used an Eye to destroy an Ascion just before this fight, which was a lot easier to do than what we had to go through the last time we wanted to destroy an Ascion (gather a ton of aether before hand, plus sacrifice a Scion). Using the eye, while also being weakened after being used to break the barrier of Azys Lla, was still much easier than what the previous method was. Thay had a fresh fully charged eye at thier disposal, they pretty much had an unlimited supply of aether.

5&6) As stated above, He just wanted to do Halone's bidding more so than become her.

7) Like The Moose stated, this one was all the Ascion's doing, which then became thier own undoing.

Bonus) One thing I believe is that the Archbishop didn't exactly become Thordan I, but that he manifested Thordan I into the giant sword he was using, much like how Odin is not the weilder, but actually the weapon weilded.

Hope some of this helps you!