I see John revealing to us the simplicity of the fact, that it is the same Word of God which ministers to each group of persons. It is the same Word that John speaks throughout his epistles to all of the church but it is the Spirit of God that is able to minister the Word to each person at whatever stage of development they find themselves. First, he writes to the children:
1Jo:2:12: I write unto you, little children, because your sins are forgiven you for his name's sake.
Because it is essential for the newly born again to understand that they have had their sins forgiven. Without this understanding how could we have confidence to approach unto God and begin to have the proper relationship with Him. We must have this confidence in the promise of God, that if we confess our sins He is faithful and just to forgive us our sins(1Jn.1:9, Eph.3:12). It is paramount that the children comprehend that they are accepted of the Father not on their own merit but on the merits of Christ's work; having their hearts established with grace;
Heb:13:9: Be not carried about with divers and strange doctrines. For it is a good thing that the heart be established with grace; not with meats, which have not profited them that have been occupied therein.
John writes to the young men in 1 Jn. 2:13 that they have overcome the wicked one. But in verse 2:14, John shows how the young men have overcome the wicked one: because ye are strong, and the word of God abideth in you. Strength here is not in relation to physical but spiritual strength (Ec:9:11-18) and it is obtained through the knowledge of God. Once we, as children, have understood that our sins have been forgiven, we must go on to grow in the Word and abide within the truth that we have revealed unto us (Jn. 15:1-8). The example of how to defeat the wicked one has been given to us when Jesus overcame the devil by skillfully quoting and obeying the Word in each occasion He was tempted (Mat. 4:1-11).
And finally John addresses the fathers (elders) as well by reminding them that they have known the Father; reiterating the same in both verses of 1 Jn. 2:13+14. This word known is used to establish that these fathers have had an intimate relationship with the Father. These elders as opposed to the children and even the young men are recognized as having been established in the faith and in the knowledge of the Word as a result of their experiences in spiritual matters. This knowing speaks to the understanding they have of the nature and ways of the Father (Jer.9:24):
Heb:5:13: For every one that useth milk is unskilful in the word of righteousness: for he is a babe. Heb:5:14: But strong meat belongeth to them that are of full age, even those who by reason of use have their senses exercised to discern both good and evil.
These verses in 1 Jn. 2:12-14 allow us to understand that there is a correlation between how in the natural there is a growth process of children to young men to fathers, so too in the spiritual there is a similar pattern. Believers must have patience to be nurtured by the Spirit of God into every age group. Peter in his epistles ministers similarly:
1Pe:2:1: Wherefore laying aside all malice, and all guile, and hypocrisies, and envies, and all evil speakings, 1Pe:2:2: As newborn babes, desire the sincere milk of the word, that ye may grow thereby: 1Pe:2:3: If so be ye have tasted that the Lord is gracious.
Peter further elaborates in 2 Pe.1:1-15 on this growth process and how we should continually remind ourselves that there is this process involved in our spiritual development.
I hope that using the scriptures quoted above helps to shed some understanding on the question
The fact that John begins with the children is because that's where a believer begins his walk (relationship)--vs.12 "because your sins are forgiven you for his name's sake"; and as a new born babe what is imperative is that a believer understand that he has been received into the family God (Gal. 4:4-8) and now is considered a "son" and now know God as Father--vs.13 "because ye have known the Father". Regarding the order of fathers first, then young men second and children last in verses 13+14, I would say It is in order of authority in the body of Christ. The elders having more experience and having their senses exercised to discern between good and evil,they know (understand) "Him (God) that is from the beginning" and have a charge to look out for the younger (1 Pet.5:1-5, Heb. 13:17). And the younger have a charge to submit themselves unto the elder (1Tim. 5:1, Heb.13:17, 1Pet.5:5). As you have noted the message to the fathers in verses 13+14 is the same, "ye have known him that is from the beginning" And the message to the young men is essentially the same "ye have overcome the wicked one"; the difference in verse 14 is John shows how the young men overcame, "men, because ye are strong, and the word of God abideth in you". As is common often the same thing is stated twice to bring emphasis to the statement. But where we seem to see things differently is in your statement:
"So, my main question here is this: Why John chose exactly these unique things for each one of the three groups (in other words, why is it specifically about sins when it comes to children, why is it specifically about the wicked one when it comes to the young men, and why is there nothing negative at all, when it comes to fathers)?"
I can only see positive things being addressed to the children: vs.12 "your sins are forgiven you for his name's sake" and vs.13 "ye have known the Father". And only positive to the young men: vs.13 "ye have overcome the wicked one" and vs.14 "ye are strong, and the word of God abideth in you, and ye have overcome the wicked one". It seems like your question is more of a statement than an inquiry. As though there is some hidden deeper meaning in looking for negatives in the address to the children and young men (where none is stated). For me, the children are introduced into the kingdom and realize their sins are forgiven. The young men continue in the way and come to realize that they are in a spiritual warfare and learn to overcome the enemy through their knowledge of the Word and trust in the grace of God. The fathers are settled (1Pet.5:10) into the way and help the younger to attain unto "the stature of the fulness of Christ" (Eph.4:11-14).