When I look at these kinds of arguments, I can always tell that they will fail even without reading the argument itself. I do have to read the argument to find the details of how they fail. That the arguments will fail, however, can be seen before reading them.
Why?
It's a general principle of logic. Perhaps it is also a general principle of life, but it is specifically, a general principle of logic. Roughly speaking, that general principle says:
- There is no such thing as free lunch.
In logical terms, the principle is that a conclusion can never yield more than is already implicit in the premises. All logic can do is rearrange the assumptions, so as to make more obvious part of what was assumed. Perhaps it is because of my background in mathematics, that this principle of logic is particularly apparent to me.
If a logical argument assumes nothing about the world, then it cannot logically derive any conclusions about the world. At best it can come up with some esoteric relation between logical connectives (the sort of thing that turns people away from mathematics).
When we see one of these arguments that purports to reach a conclusion about reality (such as that God exists), but without using any evidence, then we can immediately know that the argument smuggles in some assumptions about reality, and is using those hidden assumptions as the basis for the conclusions. So the way to examine the argument, is to look for the hidden assumptions.
Typically, in these arguments, the hidden assumptions are in the form of very general statements perhaps described as first principles. And then the argument proceeds to supposedly deduce a very specific conclusion from the very general assumption. The trouble here is that the very general statements are often only true in a very general sense - as a kind of broad average way of looking at things. If you try to pin them down to detail, then their truth becomes questionable. For example, look at premise 2 of syllogism 2 in the OP: "Everything that happens/starts has a cause." That's a good general statement that summarizes how things normally look to us. But when you get down to specifics you find that quantum physics does report spontaneous (i.e. uncaused) events. The thing to remember about logic, is that the logic requires that a premise not merely be generally true as a statement of what is typical. Rather, the logic requires that the premise be absolutely true in all details and in all relevant applications.
There's a general class of paradoxes, known as the Sorites paradoxes, that illustrate what can go wrong when you base an argument of statements of general truth that fail to be true in all details. See the wikipedia entry or the SEP entry for examples and discussion.
Looking at the arguments in the OP, they are all of this kind. The second and third use general assertions about cause as their starting assumptions. The first uses a general statement about entropy ("entropy always increases") as a starting assumption.
Even without the evidence from quantum physics, we could look at assumptions of the form "everything has a cause", and we ought to realize that this is at most a general observation and not anything that we can prove.
The entropy assumption has similar problems. For all we know, there might be processes acting in the universe that tend to reduce entropy, in such a way that they counteract the general tendency of entropy to increase. The observed expansion of the universe might be reducing entropy. Or the total universe might be infinite, so that you can always keep the entropy down by pushing it somewhere else - that's like the way that there is always room for more guests in Hilbert's Hotel.
When you see arguments that seem to assume little, but prove much, it is time to be skeptical. Always remember that there is no such thing as free lunch.