I can't speak to the western Mediterranean or Europe in these periods, but there are several good books that cover some of these periods.
One challenge you'll face is that there really isn't a good overview of everything you want all in one book. The region you are describing covers several different subfields, European prehistory, Greece/Rome, Anatolia, the Southern Levant, and Mesopotamia (just to name the big ones). So you'll probably have to read a bunch of different books and synthesize the information.
The second problem you'll run into is the fact that your interests are crossing a number of time periods. Mesolithic (or "epipaleolithic" outside of northern Europe, and is basically Stone Age hunter-gatherers), Neolithic (the beginnings of permanent settlements and agriculture), Chalcolithic (when people start to use copper), and the various subdivisions of the Bronze Age, where we start to see urbanism in full swing. Each of these periods has their own sets of specialist scholars.
That said, I do have a couple books that will get you started. Ill give you a blurb about how it fits the categories outlined above.
Yorke Rowan is one of the best current authority on the Chalcolithic period in the Southern Levant. If you're looking for an author to follow on that, he's the one. Pretty much anything he writes will fit your interests.
Another prominent figure in Chalcolithic studies is Tom Levy. As above, pretty much anything he writes will be excellent research in some of the time periods you're interested in. Specifically, I'd recommend
Levy, T. E. (1995). The archaeology of society in the Holy Land. New York, NY: Facts on File.
which is a volume edited by Levy (includes some chapters by him as well) and incorporating a number of other well-respected scholars in the field. It covers everything from paleolithic through the modern period. It's an excellent overview book, if slightly more limited in geographic scope. One thing you can do is look at the names of the authors of some of these chapters and look them up as well. They're all extremely solid scholars, some of whom I personally know and have worked with. The list of chapter authors is a bit of a who's who in Levantine scholarship. I highly recommend this book.
For Anatolia, I highly recommend:
Sagona, A. G., & Zimansky, P. E. (2009). Ancient Turkey. London: Routledge.
This is the best overview of ancient Anatolian history and archaeology that I am aware of. It really concentrates on the earlier (Neolithic) parts of Anatolian pre/proto history, and the majority of the book will cover time periods you're interested in. Anatolia is an important area in this time period, boasting very interesting sites such as Göbekli Tepe and Çatalhöyük, as well as the less popularly well-known Alacahöyük and Hacilar.
This is by no means an exhaustive list of books, there are many others that are very good. These will give you a pretty good understanding of the eastern part of the Mediterranean region for most of your time periods of interest.
Good luck!
Edit: forgot a link