petersk

I don't really think there is a connection to the use of 教 in　一教 and in religion (shukkyo) 宗教. 教 simply means to teach, like in the verb (oshieru) 教える. So if the Japanese words you mentioned, they would literally mean:
一教 - First lesson
仏教 - Lessons of Buddha
キリスト教 - Lessons of Christ
And a couple others using 教:
教育(kyouiku) - Education
調教(choukyou) - Training, schooling

So I think it would be a stretch to find any kind of connection between religion and the use of 教 in 一教.

I had to ask my wife about this one, but she has never seen 教士 before. She tells me it's not a word. The use of 教師 is used in academics. So a math or university instructor is a 教師, but a martial arts instructor or a teacher of skilled professions (master & apprentice relationships) is not a 教師.

Remember, many words may have several meanings. Just because a word is used in two contexts, it doesn't mean that there is a connection between those two things. It's almost like saying there is a connection between the sweet sticky yellow substance made by bees, and my wife just because I call them both "honey."