The following three books are, I believe, the most significant of the earlier treatments of new math, and I suspect you can find much in them that will direct you towards literature for your questions #1 and #3. Also, since it is very likely that at least one of these 3 books will be cited by any reasonably researched publication, you can google their titles for many other sources of secondary literature (which then should direct you towards more literature, for all of your questions).
Of the following three books,  is freely available on the internet and has a 24-page bibliography of 346 items. I’ve looked through  quite a lot, and further below I’ve included some notes that I have on  (table of contents and text of its introduction). I haven’t looked at  in several years and I don’t have any notes to myself about it, but  is available in most any U.S. college or university library. As far as I can recall, I have never looked at .
 Robert W. Hayden, A History of the "New Math" Movement in the United States, Ph.D. Dissertation (under William Brown Rudolph), Iowa State University, 1981, v + 271 - 2 pages. [Note: The last numbered page of this dissertation is labeled "271". However, in this dissertation there are two instances where a single sheet is labeled with two page numbers---a single sheet labeled "222-223" and a single sheet labeled "263-264".]
 Philip S. Jones (editor), History of Mathematics Education in the United States and Canada, 32nd Yearbook, National Council of Teachers of Mathematics. review by Anthony V. Piccolino of the 2002 2nd printing
 William Wooton, SMSG. The Making of a Curriculum, Yale University Press, 1965. review by Harry M. Gehman in Science
TABLE OF CONTENTS of 
I. Preface (pp. 1-4).
II. The Mathematical Background of the "New Math" (pp. 5-46). subtitled: A. Introduction (pp. 5-7); B. The Foundations of Geometry and the Reformation of Analysis (pp. 7-17); C. The Development of Set Theory (pp. 17-20); D. Non-Euclidean Geometry (pp. 20-24); E. Modern Abstract Algebra (pp. 24-27); F. The Spread of Modern Mathematics (pp. 27-36); G. The Impact of Modern Mathematics (pp. 36-45); H. Conclusion (pp. 45-46).
III. Educational Reform Prior to the "New Math" (pp. 47-71).
IV. The Second World War and Mathematics Education (pp. 72-99). subtitled: A. The Importance of Mathematics to Society (pp. 72-74); B. New Uses for Mathematics (pp. 74-75); C. The Impact of the War on Applied Mathematics (pp. 76-80); D. The Impact of the War on Society's Support of Mathematics (pp. 80-83); E. The Impact of the War on School Mathematics (pp. 83-87); F. The Impact of the War on College Mathematics (pp. 87-99).
V. Secondary School "New Math" (pp. 100-172). subtitled: A. The University of Illinois Committee on School Mathematics (UICSM) (pp. 100-107); B. The University of Maryland Mathematics Project (UMMaP) (pp. 107-109); C. The Commission on Mathematics of the College Entrance Examination Board (CEEB) (pp. 109-116); D. Sputnik (pp. 116-120); E. The School Mathematics Study Group (SMSG) (pp. 120-139); F. Other Programs (pp. 139-143); G. Dissemination (pp. 143-157); H. Reaction (pp. 157-172).
VI. Elementary School "New Math" (pp. 173-234). subtitled: A. The Problem of Teacher Training (pp. 173-176); B. The Early Elementary School "New Math" Programs" (pp. 176-189); C. Training Teachers to Teach the "New Math" (pp. 190-202); D. Dissemination (pp. 202-206); E. The Conflict Between Elementary School "New Math" and the Progressive Tradition in Education (pp. 207-222/223); F. The Neo-progressives (pp. 222/223-234).
VII. Conclusion (pp. 235-245).
VIII. References (pp. 246-269; 346 items listed).
IX. Acknowledgments (pp. 270-271).
I. Preface of  (pp. 1-4): [underlining in the original indicated by italics here]
The 1950s and 1960s saw a major upheaval in the content and viewpoint of school mathematics. The changes that took place during those years were part of a coherent movement to reform mathematics education. The various reforms advocated by the leaders of this movement became known collectively as the "new math." The present work is a history of the "new math" movement in the United States.
This movement brought about change in the school mathematics curriculum on a scale and at a rate unknown before---or since. The 1970s and 1980s have seen a drift in the opposite direction. For example, there have been demands to go "back to the basics." While it is not always clear what is basic, such demands clearly challenge recent reforms and call for a return to a more traditional curriculum. Unfortunately, such reaction against the "new math" has often caused us to lose sight of its entirely valid criticisms of "old math." What is needed is not to go back, but to go forward more wisely. Part of that wisdom can come from examining the history of the "new math."
The "new math" did more than simply criticize the old. It also provided answers to some of the problems of mathematics education. It did not answer every question, nor was every question that it answered correctly, but it did give some answers. These answers can be relevant to the educational problems of today. For example, while writing this history, the author was teaching a remedial college mathematics course in beginning algebra. The chosen text was written as if the "new math" had never existed. A successful effort was made to have the text changed the next time the course was offered. The new text had as one of its coauthors a leader in the "new math" movement. The new text was not a "new math" book, but its authors had learned from the "new math," and what they learned made theirs a much better text. One of the purposes of this history is to prevent the lessons learned in the "new math" experience from being lost. Another purpose is to keep alive an awareness of basic issues in mathematics education that were raised by the "new math."
There have been no previous attempts to write a history of the "new math." It is true that many articles, pamphlets, and books explaining "new math" appeared during the 1960s. These often contained a sketch of the historical background of the movement. However, none of these sketches pretended to be serious scholarly studies. They represented instead the impressions of those involved in the movement in the nature of its roots.
The one serious scholarly work that treats the "new math" is the Thirty-second Yearbook of the National Council of Teachers of Mathematics, A History of Mathematics Education in the United States and Canada (1970). Although this work is well-documented, it naturally only gives a part of its attention to the "new math." In addition, it puts too much emphasis on what was said about mathematics education, by various experts and commissions, and too little emphasis on what was done.
A final earlier work relevant to the present study is a history of the largest "new math" group written by one of its members (Wooton, 1965). While invaluable in its own area, the book is far too narrow to give an adequate appreciation of the entire movement and its origins.
In writing the present work, an attempt has been made to make it as useful as possible. This attempt has influenced the final work in two main ways. First, to make it readable by as wide an audience as possible, explanations of the mathematical background of the "new math" are given in terms understandable to the educated layman. Second, the most space has been devoted to topics that have not been adequately covered in other works. Thus, the activities of the School Mathematics Study Group are sketched only with enough fullness to maintain continuity, since a detailed history of this group is available (Wooton, 1965). On the other hand, the influence of World War II and of educators and educational philosophy on the "new math" have been largely neglected. Here, they are treated in some detail. Nearly every discussion of the "new math" includes mention of its roots in changes that took place in mathematics in the nineteenth century. Less common is an explanation of those changes, and a description of how those changes gradually diffused over the span of a century to influence the school curriculum. Both of these omissions are corrected here. Our story begins with an account of the effect that mathematicians and changes in mathematics had on the "new math."