クビにする means "to fire someone" as you said. It is rarely used as other meanings.
The sentence before you refer to is 朝一のアラームほど許せない輩はいないでしょ. 輩 is used for a person, not for objects. ヤツ is the same as 輩. So this lyricist regards alarm clocks as humans. That is "personification".
Daijirin (三省堂 スーパー大辞林) lists the meaning of 首にする as:
(1) 解雇する (to fire someone)
(2) 首を切る (to behead someone)
Maybe that second one fits better in the song. Then again, maybe it just means that they dislike the alarm clocks so much that they would just fire whoever made them.
For future readers who might be looking for answer, this is what you need:
It's a grammar: てみせる, which means will definitely do ~
I'll never fail to meet your expectations.
Also check out this answer: てみせる
I haven't looked up dictionaries, but inferring from the Japanese animations I have watched, こなす is something like doing something thoroughly and successfully. The most common form I have heard is 使いこなす (to use something in its full capability, or something like that). Never knew that it had a meaning like "mature"; never heard that kind of usage ...
Since I can't add a comment, I'll add to the answer above.
Context is important
The original line came from a thought by a character in the book (at least from the limited snippet I could find via Google先生):
The principal had thought that it would be useful to remove shame from the ...
You found wrong one. In dictionaries, 文語 does not stand for "literary" or "written", but Classical Japanese. The true one you are looking for is less likely found in dictionaries, because it is a colloquial conjugation.
The legitimate descendant of the "old" 持たす is 持たせる (which conjugates in ichidan), and ...