Love your icon, Mr Bean ... just don't park you car in the wrong place at the military base ;-)
"You ask where the piece of toast at breakfast came from. You could trace it back to the bakery, back to the wheat farm, back to the farmer who sowed the wheat, back to the parents of the farmer who brought the farmer into the world. Those parents had parents, etc. Eventually there has to be uncaused Causer -- whom we know as God."
Aristotle noted 4 different kinds of causes ...
Material Cause = matter in general (matter without form, so unknown to modern science)
Formal Cause = structure in general (aka physics and chemistry)
Efficient Cause = how something comes about (aka energy and process)
Final Cause = only applies to sentient beings, why something comes about
Aristotle also invented the "unmoved mover" aka the impersonal principle, which substitutes for G-d
Modern science and atheists don't accept Final Cause ... for Aristotle it is teleological (you want to go to Disney World, so Disney World causes you to go on vacation). Aristotle had a problem with the chicken-egg conundrum. He opted for the future egg causes the mother hen, he was thinking of intention. But modern science has a problem, called Quantum Mechanics ... which throws a spanner into this ... while there is little evidence that the future causes the past, even if we accept that the mother hen causes the future egg (more common sensical) we have to ignore intention (which Aristotle considered so important). The problem is that Classical Mechanics assumes that the observer has no impact on what is observed ... and Quantum Mechanics says that when a scientist does an observation or sets up an experiment ... that this changes everything (but in statistically predictable ways). So the intention of the scientist, still sneaks back in.
"Likewise the universe, being an effect that we can easily see, must have a cause. The alternative would be to say that it "just there" for eternity, which would only raise other questions. Why is it the way it is? Etc."
That which exists now, has a cause in the past, even if it is unchanged? All things change, except for physical law (which ideally is unchanged and universal in space as well). So what does change, does have a cause. But what is unchanged doesn't have a cause, it just is. But the question of how change happens is answered by science. And science uncovers the unchanging (Heraclitus was almost right). Even the 3.5K radiation form the Big Bang is slowly changing, under rules that are unchanging. We don't actually know what happened before that ... we speculate ... and people take the speculations of science too seriously. If we ask why, we are asking about sentience. If I am typing, you can ask why ... if a book falls off my shelf, you don't ask why (nobody pushed it) you ask how, and the answer is gravity (but that as you point out, still begs the question).
"Even if there was a "big bang," that too would need a cause. To say it "just happened" isn't very satisfying; it doesn't make sense."
Correct, temporality makes sense, because we are temporal sentients ... eternity doesn't make sense, in spite of the efforts of theologians.
"Things "just don't happen." If things just happen without reason or a cause, then the universe would be incomprehensible. There would be no order to it. Science itself would be impossible."
People see patterns in clouds, and the romantic see a dragon in the sky. But that pattern isn't real (aside from the human imagination). Some patterns we call real, because while they depend on the human imagination, they are objective, not just subjective. That is the scientific method, to separate the objective from the subjective (peer review, re-observation, repeat of experiment, tendency to measurable quantification).
Personally, I do take the position that science is man-made, not a Platonic Ideal ... but then I am empirical like Aristotle, not mathematical like Plato. I don't think reality is rational ... only that some aspects of human experience are objective, but mostly subjective ... that chaos is more common than order, illogic more common than logic. For me, ultimately the universe (as an idea of humans) is incomprehensible.
Things "just happen" all the time (in the sense that most often no sentience is involved, human or animal). Again, that confusion is due to misapplication of Final Cause.