Yes, there are non-deterministic parts in physics. For example atomic decay. While we can measure and work with half-life times for large amounts of radioactive atoms, the decay of a single, individual atom is unpredictable. So in a way, you can get your desired dose of vagueness by controlling how many atoms you monitor. The less, the more.

Or another example from the same field: There are atoms for which we believe they are stable, although they theoretically could decay. But we never observed it. So maybe they are in fact stable, or maybe they decay just slower than we have time. Or only when we don't look. Examples would be Helium-4 or Lead-208.

I also like the idea, inspired by Douglas Adams, that the universe itself could be a weird and random fluctuation, which just happens to behave as if it was a predictable, rationally conceivable thing. That actually, it's all a random chain of junk events, and we're fooled into spottings some patterns. This apparence could last forever or vanish the very next moment, who knows. Maybe it's all just correlation and there is zero causation. As far as I know, we'll never be able to tell. So fundamentally, all of it is a vague guess, supported by mountains of lucky evidence.

(Edit: Author name corrected)

Disclaimer: I'm not a physicist, but I am a scientist. Science as a whole is usually taught in school as though we already know everything there is to know. That's not really accurate.

Science is really sort of a black box system. We know that if you do this particular thing at this particular time, then we get this particular response. Why does that response happen? Nobody really knows. There's a lot of "vague" or unknown things in all of science, physics included. And to be clear, that's not invalidating science. Most of the time, just knowing that we'll get a consistent response is enough for us to build cool technologies.

One of the strangest things I've heard about in physics is the quantum eraser experiment, and as far as I'm aware, to this day nobody really knows why it happens. PBS Spacetime did a cool video on it: https://youtu.be/8ORLN_KwAgs?si=XqjFEjDfmnZX31Mn

Nothing is really definite. The right word to use would be consistent. We dont know the larger or the smaller picture, just that the small pocket of all physics we know is related in a certain way in a comprehensible manner.

What do you mean exactly? This question is vague... :D

One of the first things you learn in college-level science is accuracy and precision. A measurement can have a degree of correctness and a degree of exactness about the value. For example a sensor may get the wrong reading 3% of the time. When you have a big pile of readings, you don’t always have the time to validate them all. So, there is some uncertainty that you accept. The same sensor may only be able to give you an accurate reading down to a specific decimal point, which is expressed as precision. Anything less is given as a range in which error exists. These ideas are important, because when you do calculations based on those readings, you have to take the error with you. There may be a point where the value you reach is overshadowed by the magnitude of the error.

I've read that all math breaks down as you approach the big bang. I'm not educated enough in math to understand how, or why, but apparently they cannot mathematically understand the origin of the universe.

No, physics is never vague. Some problems are currently computationally intractable beyond a specific level of accuracy (like predicting the weather more than 2 weeks out), and some questions we do not know answers yet but expect to find answers to in the future (like why did the big bang happen). But there is never an element of the mysterious or spiritual or "man can never know".

Popular science physics often gets mysterious, but that is a failure of popularization. Like the observer effect in quantum physics, which is often depicted as a literal eyeball watching the photons go through a double slit (?!). This may cause a naive viewer to mistakenly start thinking there is something special or unique about the human eyeball and the human consciousness that physics cannot explain. It gets even worse - one of the most popular double slit videos on youtube for example is "Dr. Quantum explains the double slit experiment" (not even gonna link it) which is not only laughably misleading, but upon closer examination was produced by a literal UFO cult, and they surreptitiously used the video to funnel more members.

Or the "delayed choice quantum eraser experiment" which confounded me for years ("What's that? I personally can make a choice now that retroactively changes events that have already happened in the past? I have magical time powers?"), until I grew tired of being bamboozled by all its popular science depictions and dug up the actual original research paper on which it is based. Surprise! Having read the paper I have now understood exactly how the experiment works and why the result is sensible and not mysterious at all and that I don't have magical powers. Sabine Hossenfelder video on youtube debunking the delayed-choice quantum eraser was the first and so far one of only two videos I have seen in the years since that have also used the actual paper. This has immediately made me respect her, regardless of all the controversy she has accumulated before or since.