@heidigfx Happy to get into the specifics. You ready‽ Let's dig in!
Illustration Size: This really means the size of the book. That is determined by your editor and art director. In my experience illustrator has little to no input on this. (although I'm sure there are exceptions)
Illustration Number: While the page count of the book is set by the publisher, typically the number of images is initially determined by the illustrator as they go through and create the rough dummy of the book. That number may increase or decrease through the editorial process as the illustrator along with the editor and art director decide (for example) to make a spread with two full page illustrations into a single full spread or break up an image into a series of sequential spots etc. (My own habit is to over illustrate in the rough sketch stage and then progressively eliminate images that are unnecessary or overly complicated)
I should note that I am thinking of trade picture books here. If you are creating a book for the educational market the art direction tends to be more specific and heavy handed.
Cover Illustration: (Again, thinking trade picture books) Cover is always included in the advance against royalty payment. The choice to make the cover illustration front cover only or a wrap around illustration is something determined in conjunction with the editor and art director.
Thumbnailing: Do as many rounds as you need to make the project successful (i.e. TONS!!). Only show the "rough" dummy to the editor.
Revisions: I incorporate editoral suggestions after submitting the rough dummy. I will incorporate editorial suggestions after submitting the finished sketch dummy. And that's it.
Character, Set, Prop Designs: It is not uncommon to create a sample character design (usually of only the main character(s)) and even a full color sample illustration of what the illustrator imagines the book will look like at the very start of a project (after the contract has been signed -- I don't advocate working on "spec"). This gets everyone on the same page visually. Everything else can be approved or commented on when evaluating the rough dummy and the finished sketch dummy.
Payment of advance against royalties: Broken up into thirds. 1/3 on signing the contract. 1/3 on submitting the finished sketch dummy. 1/3 on submitting the final art. I don't submit an invoice for trade picture books. Typically the editor triggers that payment within the publishing house. If needed I will e-mail a reminder. You certainly could submit an invoice if needed.
Contracts: I heartily recommend that you familiarize yourself with book publishing contracts, but it is more likely that the publisher has their standard (also referred to as "boilerplate" contracts). These are (by their nature as contracts) negotiable, but it's unlikely that you'll be submitting your own book publishing contract. That said, SCBWI has a contract available:
I like the one available in the Graphic Artist Guild's Pricing and Ethical Guidelines:
I particularly like Tad Crawfords Business and Legal forms for Illustrators:
as I find he does the best job of not only providing a contract but clearly breaking down what each clause of the contract means in plain english.