Which books exist that cover the history of analytic philosophy? Which papers are integral to the field and as such are a must-read?
The first four chapters of The Oxford Handbook of The History of Analytic Philosophy are available for free from that webpage. The four chapters are:
These are all incredibly comprehensive articles and do a great job condensing the information into one source. You would probably be well off using this as a guide and then following the bibliography and reading primary sources and articles if you want to really understand the material.
For additional sources, if you want them, the prominent analytic philosopher Scott Soames as taken it upon himself to author multiple multi-volume series on this subject. So far there are:
The Philosophical Analysis in the Twentieth Century series is only those two volumes and provides a very thorough introduction to the subject and all of its history, while also positing some original claims about how the history and the subject should be interpreted as a philosophical field in and of itself. The The Analytic Tradition in Philosophy series is a much more in depth series that covers a lot of fine grain details starting all the way from the beginning and is planned to be a five-volume series. Only two have been published so far and I believe the second volume makes it only to the 1930's.
Any of the above linked books should serve as a great introduction to the topic.
In terms of what papers are a “must read”, you would probably want to be familiar with all of the biggest papers/books by the biggest names in the field.
Gottlob Frege’s Sense and Reference (There is probably some version of it for free online somewhere.)
Bertrand Russell’s On Denoting
G.E. Moore’s Principia Ethica
Ludwig Wittgenstein’s Tractatus Logico-Philosophicus
Rudolph Carnap’s The Logical Structure of the World
W. V. O. Quine’s Two Dogmas of Empiricism
Saul Kripke’s Naming and Necessity
These are just some examples of papers that have had the biggest impact on the field by some of the field’s leading figures. They with undoubtably come up as you are reading over a history of the field, and a majority of the world done by later philosophers references either these papers themselves or responses to these papers, especially in the philosophy of language. I’ve seen it stated by many people that Quine’s Two Dogmas is the most influential philosophical paper that came out of the 20th century, but I currently don’t have a link to a citation on that so take that statement as anecdotal conjecture; however, it is without a doubt one of the most.
You will definitely want to read Hans-Johann Glock's What is Analytic Philosophy? (2008), where the author addresses the question by approaching it from various angles: historical, methodological, topical, ideological, ethical/political and geographical. It is not only a great, informative read, but also contains a formidable bibliography addressing your second question.