So the thesis of "Why the West Rules" is basically it had beneficial geography compared to the East.

You are right to be uncomfortable with the book, it is just another pseudo-geographical "historical" book which seeks to revise actual world history through the lens of a few geographic factors. Its just essentially a re-write of Guns, Germs and Steel.

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"1. How correct is he when he puts Mesopotamia as part of western core? "

The whole debate of how Western the Middle East is, is a big one. Historically, Middle East was regarded among traditional historians as Western until about 9/11. For instance, if you can find a history book on religions written prior to 2000, they refer to Islam, Christianity and Judaism as Western religions and Western societies. They refer to Hinduism, Taoism, Buddhism and so on as Eastern. Historians have a more nuanced and better understanding of Eastern and Western historical civilizations. It is not so much that you have a strict East or strict West, there was lots of trade, a continuum. All kinds of cross interactions, peoples moving in ways we did not understand or acknowledge before. When someone says, "Mesopotamia is Western", the question that should be asked is what is Western, who is Western, what is the cut off, who determines Westernest, what is the criteria? You will find more often than not, the people using these terms cannot answer this, they use empty adjectives they struggle to explain. After 9/11 a big movement emerged in much of Europe and USA among conservatives that Islam is Eastern and all the Islamic countries are now Eastern.

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2. "Does the west (let's say Europe of today and European nations in new world) really came about from one civilizational core unlike Asia with different cores in China, South Asia and middle east? "

The question is a bit convoluted and I am unclear as to its meaning, I'm going to need you to clarify some parts to me. A. What is a European Nation in the New World B. I know of no European country in the New World, are you referring to European colonies in the New World like St. Pierre and Miquelon, Aruba, Caymans, French Guyana etc?

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If we define the West for this purpose as just being Europe, in absence of your answer, Europe has a complex and complicated history. I'd say probably Greece and Rome in combination with the surrounding Asian and African nations all lead to the establishment of those civilizations in Europe, it was not a straight line from Mesopotamia to European civilization. Then of course you have levels of civilization.

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3."He postulates that while opposite could had happened in 19th century i.e. Chinese ships bombarding London, it was impossible that similar thing could had happened with a south-east Asian, eastern Saharan or south Asian ship. I think the author is not going to explain why as in the difference between these civilizational core rather than just Chinese and western, so can I get suggestion for some other good resources? "

He isn't going to explain why because it is just an empty opinion with no supporting facts. Had the South East Asians or East Saharan or South Asians had an interest in bombing London, they would have spent their money on doing so.

My view of Morris' book is a lot more positive than u/White_Mlungu_Capital's, so I'll give a response from a more charitable angle. I'll start by answering your numbered questions and then give a general defense of "this type of book" and its place in historical scholarship. I should state upfront that I'm an enthusiast, not a professional, and that my expertise is mainly in early China.

1. Definitions of "Western" and "Eastern" are necessarily subjective and tend to also be politically or ideologically loaded, shifting arbitrarily to match the commentator's political or social views. To avoid (or at least mitigate) this problem, Morris identifies "Western" and "Eastern" with the early emergence of agriculture in the Middle East and central China, respectively, and does his best to stick with it. To continuously treat Europe and the Middle East as part of "the West" is consistent with the definitions and objectives he sets out in his book; whether this definition is actually the best or most appropriate one in general is an open question, and it certainly isn't used much outside of this particular book, if at all. But within the scope of Morris' project I would argue its the most reasonable and consistent approach that he could take.
2. Yes, this is basically accurate. European agriculture started with the outward migrations of Middle Eastern farmers, and all evidence points to independent domestication of different animals and food crops in the other "cores" identified by Morris. It is important to note that over time, these "cores" all exchanged crops, animals and techniques so that these differences became less and less meaningful; and of course each "core" was both internally divided and interlinked with the others to varying degrees.
3. Morris may be overplaying his hand here, and there's certainly no deterministic reason why an industrial power couldn't have risen first in Mexico or Bengal or some other region, as Morris himself acknowledges. Still, identifying East Asia as the closest "runner-up" and most likely alternative has some basis- this shows most clearly in the chapters that deal with the Song Chinese "commercial revolution." Historical economics is a fast-moving field and the "Great Divergence" debate which he describes in the introduction is by no means settled- to get a better grip on this, I would seek out economic histories of various regions that catch your attention as well as "Great Divergence" works (Pomeranz, Frank and Goldstone, referenced in this book, are good representatives of some of the schools of thought on this subject).
4. There is a growing literature on "comparative empires" that I strongly recommend looking into- it has given a great deal of focus to the Roman and Han empires (because they were contemporary, similar to each other in a number of regards, yet had minimal contact with each other) but the field is diversifying and new works are coming out rapidly that cut across more regions and time periods. The beating heart of this field is at Stanford, where Morris is a professor of Classics, by the way.

Now comes the part where I have to disagree sharply with u/White_Mlungu_Capital, and possibly with general opinion on this sub. I will commit the heresy of heresies and say that, despite its numerous and well-documented flaws, Guns, Germs and Steel laid out a valid and worthwhile thesis ("geographic and ecological advantages mostly account for the historical dominance of Eurasia over the Americas, Oceania and Subsaharan Africa") and has made a positive overall contribution to historical understanding, especially outside of the academy. By extension, Why The West Rules- For Now makes a meaningful refinement of Diamond's thesis ("within Eurasia, geography and ecology further advantaged the Near East and adjacent regions over the others, although not to the extent that it was the sole determinant of development").

These are general works, which cover expansive topics far too vast for any one author to ever be a genuine expert on, and are further constrained by the need to make their arguments appreciable to laypeople who are not conference-attending, journal-reading historical scholars. Why the West Rules- For Now covers Eurasian prehistory and history from the early hominids to the present day in just about 600 pages- how could a work like this be anything but overcondensed, oversimplified, and full of the misunderstandings and misconceptions of the half-expert? Specialists should be able to poke holes and catch inconsistencies in works like these. The real question is whether or not these books still have value in spite of their limitations. I argue that they do, partly because of their ability to raise new research questions for the specialists to dive into but also because of their ability to communicate general historical concepts or schema to general audiences, who will develop their own ideas with or without the input of scholars. The discipline needs both esoteric and exoteric works if it wants to remain healthy.