Perhaps because they're followed by a vowel? That seems to be the obvious difference between the first set of words and the two exceptions that you've given.

I've never heard anyone use "archconfraternity." So, you probably do not have to remember that one. It could be the vowel. It could be words that stopped off in French or Italian before coming to English rather than coming directly from Latin to English? As you've undoubtedly noticed, English spelling and pronunciation has a lot rules and a lot of exceptions. Even native English speakers make mistakes when they come across an unfamiliar word and try to figure out what it sounds like based on how it is written.

I don't know for sure. I think Su.Ki's answer is right because the "arch" in these words could be a contraction from the Greek "arche / archi", to avoid two vowel sounds together. Other examples with a /k/ sound that I can think of are WHOLE words that come direct from archipelago, archetype, architrave. However, in words like "arch(-)fiend", arch(-)enemy, the prefix "arch" has been added on to the front of an existing word. The hyphen that you sometimes see in some of these words indicates this.