I think the 7 is quite obvious, as that is the number of heavenly completion, as the earthly completion number is 10 (10/10). That would be where there are 7 female Valar and 7 male Valar ("heavenly beings", if you can call them that, although Melkor is the odd one out, as he was kicked out of the gang for not being cool)
I remember reading somewhere that the number 9 is for symbolizing re-birth, but I'm not into all these symbols using planets etc, just doesn't interest me. But, the nine Nazg’l died and were all "brought back" like a re-birth to do the will of Sauron. The fellowship were formed so as to counter the "re-birth" and make an equal stand against them. So they were "re-birthed" themselves into the fellowship.
But this is all what I don't know completely, but I think that some points make some logical sense
Anyway, does anyone know what is the significance of the specific numbers: 7 dwarves, 3 elves, and 9 men?
There were seven Dwarven lords, 3 Elven Kings and 9 Great Kings of Men. Hence, Gwaith-i-M’rdain in Ost-in-Edhil made seven rings for the Dwarves, three for the Elves and nine for Men.
That's all. No deeper meaning behind it. Sometimes a cigar is just a banana.
The 'sixteen' were later given to other races by Sauron, after he took them by force (though the Dwarves say one of the 'Seven' was given to them by the Elves). The M’rdain did not set out to make a certain number of Rings for each race; the 'numbering' came about as a result of Sauron (though he never touched the Three of course).
Anyway, we learn in RC:83-4 that "In the earliest complete version of the Ringverse there are nine Rings 'for the Elven-kings under moon and star' and three 'for Mortal Men that wander far' (RS:269, no.14). Another had twelve Rings for Men, nine for the Dwarves, and three for the Elves. The numbers and powers of the Rings varied as the story developed." (My emphasis) "The term Elven-kings seems to be merely poetic; later in this chapter Gandalf refers, more appropriately, to 'the Elf-lords', i.e. the leaders of the Elves in Middle-earth."
So I think there isn't any number symbolism here, though you can start ruminating about the divine properties of the number Three, combined with the number of elements (Four) to make Twelve; and of course Seven and Thirteen have been closely linked with Dwarves in Tolkien's legendarium (Seven Fathers and Thirteen original Dwarves (six each with their female counterpart, and Durin who "walked alone").
I'll add an interesting letter in which Tolkien responded: 'There is no 'symbolism' or conscious allegory in my story. Allegory of that sort 'five wizards = five senses' is wholly foreign to my way of thinking. There were five wizards and that is just a unique part of history.' JRRT 1957 Letters
Make of that what you (anyone) will. This letter then moves to 'applicability'