With respect the fundamental mistake is yours. The cosmological argument tries to straddle both experience and pure reason, with the unsatisfactory result that it can neither be one thing nor the other. And instead of presenting a self-evident truth, an ontological argument, which cannot be denied without self-contraction, the cosmological argument tries to argue backwards, by inference, from the world of experience. An unfortunate consequence of this is the unintended claim that all worlds are as ours; so that what is true of our world, for example the phenomenon of cause and effect, is also true of God. So, on this account God is absurdly dependent upon the material world for his existence and is therefore part of it. And thus if the material world need not be, the same can be said of God.
The cosmological argument has two parts. The cosmology part which is natural is examined by natural law. Philosophy bridges that natural understanding with the supernatural through principles that have no natural dependence. It is obviously a grand and valid argument because the most brilliant among us (even those that reject it) have written on it for thousands of years. It can't be snuffed out by misapplied technicality by a poster in a forum. It will be just as profound and discussed a thousand years after we are dust.
A first cause does not necessarily exist in order to explain a series of causes and their effects!
Yes it does, and it does so necessarily. An impossible infinite regression of causation will never produce anything..
Yes, please state the argument formally (instead of arguing from authority), for then it can be formally rebutted.
The original by plato, etc... is a prime mover theory.
First Kalam:
1.Everything that begins to exist must have a cause of its existence.
2.The universe began to exist.
3.Therefore, the universe has a cause of its existence.
Second Aquinas: A version of the cosmological argument could be stated as follows:
1.Every finite and contingent being has a cause.
2.A causal loop cannot exist.
3.A causal chain cannot be of infinite length.
4.Therefore, a First Cause (or something that is not an effect) must exist.
Then Craig:
According to the argument, the existence of the Universe requires an explanation, and the creation of the Universe by a First Cause, generally assumed to be God, is that explanation.
In light of the Big Bang theory, a stylized version of argument has emerged (sometimes called the Kalam cosmological argument, the following form of which was created by Al-Ghazali and then strongly supported by William Lane Craig):[9]
1.Whatever begins to exist has a cause.
2.The Universe began to exist.
3.Therefore, the Universe had a cause.
Finally Leibniz:
The basic Leibnizian argument has the following steps:
(1) Every contingent fact has an explanation.
(2) There is a contingent fact that includes all other contingent facts.
(3) Therefore, there is an explanation of this fact.
(4) This explanation must involve a necessary being.
(5) This necessary being is God.
That last step is the only one open for debate. It is by far theh best conclusion available but is not made necessary by the previous steps.