Alright, it took longer than expected. I wasn't able to give full attention to all the arguments of the book, but I think I got the main arguments and counter-arguments.
So the thesis of the book "Five Proofs" is as it says in the intro---the real debate is not between atheism and theism---by trying to show that God, as accepted by theists, definitely exists.
By God, the book means an entity with certain qualities, which I'll divide into two sets:
* impersonal: simplicity, immutability, immateriality, incorporeality, eternity, necessity
* personal: will, omnipotence, omniscience, perfect goodness, love
My main criticism of the five proofs is that, even if accepted, they can only used to demonstrate the impersonal qualities above. The book bolts on the personal qualities with thin arguments that don't follow from the proofs. And this has been my observation in many such discussions, that the meaning of the word "God" switches mid-conversation to fit the conclusion.
An impersonal "God" is actually compatible with naturalism, because here "God" is just a word being used to describe the ultimate nature of reality. So whatever the ultimate nature of reality is, if one wants to call it God, that's totally fine. To differentiate my perspective, I'll call that ultimate reality the Theory of Everything (ToE) instead. ToE has all the impersonal qualities, ToE is what sustains the universe, ToE is eternal and immutable, etc.
* ToE is what sustains the hierarchical series at each moment.
* ToE is base part out of all other composite parts of made of.
* I don't see why "there must be a necessarily existing intellect which grasps all of the logical relationships between all propositions". Truth and logic just are. They don't need to be grasped by anything to exist. However, if one wants to call this collection God/ToE, that's fine. But it doesn't follow that such a God is omniscient (as there isn't any will that "understands" in such a collection).
* I don't really see how the Thomistic Proof is different from the Aristotelian Proof, but in any case, I think ToE can be the essence of this universe's existence.
* "There must be at least one necessary being, to explain why any contingent things exist at all". Yes, the First Cause, but ToE and not a personal God.
ToE/God, with the impersonal qualities, can be the final conclusion from all the 5 proofs. But theists need "God" to be something extra. They need God to have the above personal qualities as well. And that cannot be shown with the proofs. Because the reality we observe is consistent with an impersonal ToE creating and sustaining it. ToE doesn't need to have a will, or to be perfectly good, or to love its creation. These qualities don't follow from the proofs at all. So personal qualities can only be accepted if one accepts some revelation to be true. But that is outside the scope of this discussion. All I want to show is that the thesis of the book is incorrect and atheism is back on the table.
To reiterate, something cannot come out of nothing. Cogito, ergo sum. Something (I) exists. Hence there is some brute-force First Cause for this something. Theists call this First Cause God. Atheists can call this ToE (say). I claim that God = ToE at this point. Theists go one step further to give personal qualities to God, who willed this universe into existence (but could have chosen not to) and cares about its constituents, including humans, whose prayers and actions He listens to and judges, and has revealed himself at least once (if not more). Atheists reject this second part. I instead adopt a naturalist viewpoint modulated by Bayesian Reasoning. If I find some strong evidence for a personal God, I'll of course have to change my mind and become a theist.
