Who says he ever had to admit how strong his ability really was? People LIE. They've frequently referred to the people who are Level 5's as a total number. In the schools we've been introduced to, Mikoto is the only Level 5 that people know about perhaps because she is so...flashy. (*ba dum crash*) Accelerator was known to Mikoto and MISAKA, but Touma had obviously never heard of him, and the incident was fairly isolated from most of the characters. They frequently say, "One of the only [forgot the number] Level 5s in the city," but they never elaborate. Touma certainly can't name all of them, and it seems more like only those Level 5s know of each other by reputation. I don't know for sure if he is one, but he was boasting in just the same way that Kuroko announced Mikoto (or Mikoto herself) and Accelerator talked about himself.

In addition, who said that Espers can't be members of Necessarius? They aren't mages, just an association of those who deal with magic and Espers from the churches. They are a religious association, like Masons, Knights Templar, Opus Dei, vampire hunters or any other secret organization that is commonly abused in fiction. It just so happens that most of the members of Necessarius and Annihilatus are mages because that's what they know best.

Just because espers are science and magic is divine, doesn't mean they can't cooperate. The Catholic Church has dozens of scientists. One of them even recently decreed the possibility of alien life to exist and mused on the idea of where they stand in God's vision. Science or even science fiction and religion are not mutually exclusive.

Tsuchimikado doesn't have to know how to use magic, he can know about it just the same as Touma once did. Or Loli-sensei or even Index.

But they also slipped in the lines from Misha Kruschschev and Kanzaki about how there was some corruptive way that Espers could utilize magic.