No - anyone who tries to get you to pay a fee is not legitimate.
Publishers do not ask your age - they read your book and see if it's good enough for them to buy from you.
By writing books which are well enough written for many people who want to read it, and, hopefully, to buy it. If so the publishers who prepare books for publication will pay a share of the profits. or a percentage depending upon the book's success.
Random House and Scholastic Corp. are the two largest children's book publishers.
Check publisher's websites for their submission policies. Follow them to the letter. Read Children's Writer's And Illustrator's Market. Your public library has a copy. Never pay anyone to publish your book. Those are vanity presses.
Get the most recent edition of Writer's Market - it lists every publisher, their address, the editors, what they publish, and what they pay.
Authors do the work of writing books, editors do the work of editing them, printers do the work of printing them unless the books are ebooks, publishers go to the expense of advertising books, and everybody has to be paid for their work. People do not do these things for free. Hence, the consumer, who wants to read a book, has to pay for it. Or borrow it from the library. (Libraries also pay for the books that they have in their collections.)
The most common children's book publishers is Peace Hill Press. You can see a list of the publishers at http://en.wikipedia.org/wiki/List_of_publishers_of_children%27s_books
Yes, there are many great book publishers in the US and around the world. Some of the best publishers are Grand Central Publishing, Random House, and Harper.
After you have carefully read it and edited it to ensure there are no errors in it, you get copies made and start sending it to as many publishers as possible.
different publishers offer different amounts based on the books quality.
Legitimate publishers typically do not request money upfront from authors. Traditional publishing models involve the publisher covering the costs of producing and distributing the book, with the author receiving royalties from book sales. Be cautious of publishers that ask for money upfront, as they may be vanity presses or self-publishing companies.
Otto Frank eventually decided - at the urging of several who had read it - to publish Anne's diary. But no publishers were interested. Then the manuscript was read by Jan Romein, a famous and respected Dutch historian. Romein's review of the book in "Het Parool" immediately caused several publishers to express interest in the book, and in 1947, it was initially published by Contact of Amsterdam.