It would be my pleasure. And I likewise understand your mentality concerns, then again lately it's no longer quite as chic as being wealthy even in the US. But I digress. To ensure I do not miss anything, I will go line by line here as well.
1. Incomparable to present conditions living improvements (living in 2-story villa with penthouse and jacuzzi on the 3rd floor, pool a few steps away from the front door in the elite international community (versus living in the typical project apartment in the typical project neighborhood).
Excellent point. You may want to modify the language to state that the units are 'akin to tenements' and that the child would have his own independent and safe space at the new location.
2. Incomparable level of education (internationally acclaimed private school with 3 languages - English, French and Spanish - classes (versus the typical NYC public school in typical project neighborhood).
Extremely valuable point. Soften the very legitimate dig at public schools and point out that the child will be exposed to additional private intensive educational means at the new location.
3. Living in international upscale community (versus living in the project NYC neighborhood).
Utilize the word 'diversity' along with upscale.
4. Favorable financial position of the father.
Good, your position is valuable as the more you can do, the more the child receives.
5. Intensive social life of the child (versus virtually total isolation in New York City).
Use examples here if you can.
6. Healthy lifestyle and happy childhood with extensive outdoor activities (swimming, snorkeling, bicycling, tennis, horse riding, outdoor games, etc.) in a year round perfect weather (versus factual imprisonment in the project apartment in New York City).
This and the previous point are somewhat similar...point to 'greater opportunity'.
7. Absolute safety in the gated community located in the part of Quintana Roo state that has near zero crime rate versus dangerous living in the NYC project neighborhood.
bring statistics with you if you can.
8. Abundance of entertainment (amusement parks, water parks, thematic parks, national park, zoo, swimming with dolphins, etc) minutes away on top of all regular Western entertainment and activities (contemporary cinema complex, bowling, carting, "play-hall" the size of football field with all possible games imaginable, etc) versus once or twice in a month going to a movie theater and occasional (weather/availability of the mother's time permits) attending small concrete square called playground in NYC.
Likewise opportunity, and an excellent point.
9. Incomparable intellectual development. Mother of the child has only a basic level of education, no knowledge in literature, art, music, science or/and virtually any discipline. She has no profession and/or skills and cannot support intellectual development of the child. Father of the child is internationally acclaimed classical composer and concert pianist that was chosen among the best musicians in the world to perform at 100-years Carnegie Hall anniversary in 1991. He has the equivalent of PhD in music, degree in music synthesis, degree in music recording and extensive knowledge in literature, art, music, history, philosophy, geography as well as in many other disciplines. He also has extensive knowledge in medicine/natural medicine, released a book dedicated to the treatments of degenerative diseases and a member of the American Herbal Academy. Living with the father that can support intellectual development is in the best interest of the child.
Don't bash the mom directly, but focus on the intellectual growth that will assist the child in the future and will be in the best interest going forward.
10. Living school year in Mexico and school holidays in United States will provide extensive knowledge of two cultures, multilingual abilities and will support the development of the child.
Good point. This one may or may not be helpful but it does not harm your points. You may also state that you are not considering becoming a full ex-pat and giving up citizenship, it is simply the best atmosphere for development for the child.