Yes but...
Many of the "laws" of classical physics are actually rather conditional - they work OK in normal environments, but they tend to break down in extreme cases, like in black holes. As a result, physics has developed a Standard Model of subatomic particles, which interact in non-intuitive ways to produce the classical physics we know.
As an analogy, classical physics says "this train takes 2 hours to get from A to B". Modern physics says "this is how the wheel bearings work, and the engine, and...". If you only need to get from A to B under normal conditions, you're good with the classical model. If you've been hit by a cold snap which freezes the wheel bearings, you need to deal with the extra detail.
The Big Bang definitely qualifies as "extra detail" territory, of course. You can forget about Newtonian physics there. But the Standard Model is still holding up pretty well.
So those subatomic particles...
We can do experiments on Earth under extreme conditions in particle accelerators, and see how things behave now. Because light (or other emissions) from distant stars represents events in the distant past, we can observe them and compare those ancient cosmic events against what we observe today. (And we can do experiments with light and other radio waves to check how they work right now too.) So far, everything does seem to be to tying up.
That doesn't mean that some other thing couldn't turn up, of course - just that the observations we have so far are matching well enough. As well as we can measure, anyway. Maybe better measurements will show us something new though. And that's why experimental physics gets expensive, because "better" costs money!
And those "laws"...
In some ways, physics is still at the point of describing how things work without totally understanding it. The point of building a unified field theory is getting to a point where we do understand properly. This is still TBD though.
The definition of a "theory" is that it has to match evidence. So the problem isn't just putting some maths together - it also needs you to design experiments which will show (or not show!) the universe following your idea and not doing something else. Up to now, we haven't been able to do that. (See "expensive"!)
In short...
The only answer so far is "probably", with boundary values for where this could be different and how much by. But those boundary values represent a very small amount of the universe by space and time.