One of my main specialities is musical composition. This means that I can fit the song (in fact any song) to match your vocal range. You just need to let me know what song(s) you would like to sing!

Singing is something you can train. It depends on how much someone has been exposed to music (any genre) and their interests in it, however there are situations where children do not have the ear or ability to sing, no matter how much time and money you throw at the quest. In this case, I do not want to take your hard earned money for a goal that may not be achievable and I will let you know if we think that energy and money should be spent elsewhere.
Unfortunately there are music schools that provide singing lessons for any children and are happy to put a child on stage or in public to sing when they are not ready, or with a song that does not match their vocal range. This is unfair.
Music must build a child or adult’s confidence and they must feel good about singing and feel good about making progress every week. Confidence is priceless and will carry over into everything else the student puts their mind on in life. Let me put this another way. Can everyone be an Olympic athlete? There are amateurs, and professionals. There are many more amateurs than professionals. Some people have their genetics against them. Some people have the right genetics - however they still have to train hard every day (many hours) and ensure their lifestyle is in agreement with their goals.
It depends on the level you want to get to.

It is better to stop training while your or your child’s voice is breaking because it is like training with an inconsistent instrument. The results are unpredictable and it is frustrating and disheartening. Either have a break, or we can focus on things like music theory and/or sight reading that will help him in the future. We can still do solfeggios which will teach sight singing and pitch.
In terms of how long it takes for a child’s voice to become stable, I can’t answer this. For myself, it took approximately 3 months, but every person is different.