Is there a sin on whoever uses the phrase "Allah al-Musta'an (God is the one whose help is sought)" to remain silent from commenting in discussions, and does the younger brother fall under the prohibitions for not liking this style?

"Allah is the one sought for help" is a good statement and an excellent remembrance of Allah. There is no objection to commenting with it on a specific matter, provided it is used at the appropriate time and place. Hating the remembrance of Allah Himself is an act of disbelief (kufr). However, if the hatred is directed at the speaker's manner rather than the word itself, it does not constitute disbelief. For, the natural dislike for hardship is not the same as hating the command of Allah.