Where are the properties of all cosmic objects stored?

Any implementation of memory that's ever existed uses measurable physical quantities/states to represent information, not the other way around. Also, (in classical mechanics to keep it simple) it all comes down to position, momentum, and fields. E.g "size" and "composition" is determined by the positions of particles, etc. So in a way the answer to where those properties are stored is right there.

If you don't think the universe can intrinsically have its own state, but it needs to be stored elsewhere, then there are a couple of issues. If this memory can only have the properties of the universe and no more, and can't be observed, it might as well not exist. If it has other possible states, then it's circular, and you'll need some memory one level up to store the state of the memory. So if you believe in nested simulations, I suppose your questions could be answered up to the "root" universe.

Whether there is some "external" memory that actually stores all the state in the universe is at least for now pure metaphysics. Do you think anyone on Earth could possibly answer this question today?

Size, speed, position, composition are physical properties of real objects. When you speak of memory, you're referring to either computers or memories in the brain, which are both virtual representations of real objects.

If Bostrom's simulation argument is correct, then it stands to reason that some form of memory exists to store the "physical" representation. I find that people make the mistake of thinking you would need massive storage for this. You would only need some minimal representation of the world and the physical properties would be derived based on your observation. In other words, only render what is in your field of view. If you aren't looking at the atoms, then the atoms don't exist.

https://en.wikipedia.org/wiki/Phase_space