Are the laws of physics something that exists separately from the universe or is it a description of the physical properties of the universe and objects in it?
This is more a philosophy question than a physics one, but I'll answer it here anyway. Laws of physics are a subset of mathematical statements. Mathematical statements are, in turn, a subset of abstract objects.
The ontological status of abstract objects (i.e. whether they exist, and in what way) is a contested question in philosophy. There are prominent philosophers who take opposing views on this.
The position that laws of physics are simply descriptions of what the universe does is a reasonable one and satisfies Occam’s razor. However, that still leaves an interesting philosophical question: what is it that "makes" the universe continue to obey those specific descriptions?
The laws of Science are an attempt to describe the physical universe in the most economical way possible. An extreme reductionist would say that the laws of Physics suffice to do this. The economy comes about through a few 'powerful' laws having very many consequences that can be deduced from them. The laws, though, don't exist in isolation from each other, but are embedded in a web of concepts.
Are the laws themselves, and the human reasoning associated with the laws, included in the universe that they are describing? I think that most scientists would say "no", but this is a philosophical, specifically a metaphysical, question rather than a scientific one.
The laws that actually govern the universe are of course a part of the universe. What you are getting confused about is that most people who use the phrase "laws of physics" are referring only to their hypothesized laws that they believe or have empirically verified to be accurate approximations of the actual laws. Nobody knows what the actual laws are, but we can be very sure that some models of certain physical phenomena that we consider as a part of solid mainstream science are extremely accurate in the domain of experiments that we have performed so far, such as Einstein's theory of general relativity. It may be that the universe deviates slightly from what Einstein's theory predicts (such as at the quantum level). Even if we later come to a general consensus on a more accurate theory of gravity, it can never be provably correct.
So it is true that physicists' hypotheses about the laws of physics are more or less descriptive in nature, but that has nothing to do with the real laws (which we will never have access to).
I also want to emphasize that (contrary to another answer) we definitely cannot claim that the real laws governing the universe can be expressed as statements in modern mathematics. If you know just a little about foundations of mathematics, you will know that modern mathematics is based on ZFC but there is no physically meaningful ontological justification for ZFC, so it is unclear whether every statement over ZFC has physical meaning or not, or whether real-world phenomena can be meaningfully expressed over ZFC or not. Mathematics, after all, is a sociohistorical product of the human race, not something intrinsic to the universe.
In short, not only do we not know what the real laws of reality are, but we also do not know what foundational system may be compatible with them. In fact, it appears that there is no real-world embedding of the standard mathematical conception of natural numbers even though PA yields accurate real-world predictions. I shall leave you to draw your own (philosophical) conclusions.