What things in our universe can be considered uncountable?

The fact that you always find a number in between two other numbers is not exclusive of uncountable sets, like the reals. Rational numbers are countable and have that property too. All the examples you used are countable. Both real numbers and rational numbers are not well ordered sets on their standard order (the ordering in the real line).

In physics we use uncountable sets of at most $\aleph_1$, the cardinality of the real numbers. They appear naturally in our description of space. There are some people that propose that space is best described by either a countable set, or a space of higher cardinality, such as the surreal numbers. But for now the real line works fine. Most variables in physics that are defined or related to space (likely perhaps most quantities in physics, such as forces, energy, temperature, etc) are of uncountable cardinality, because the uncountability of the real line permeates through them. But many other variables are countable, such as number of particles, etc. However I do not find myself capable enough to give you a thorough list.

Your argument for the uncountability of the real line does not work, as it would show that the rationals are uncountable.

Both the number of blades of grass and of grains of sand are finite, so they have nothing to do with infinity.

You say that reality "behaves nothing like the real line"; yet, calculus is used to put a rocket on the moon, or to do a flyby of Pluto, and a million more very real applications.

Do I claim that the reality we perceive is uncountable? Not at all. In fact, it would be impossible to test such hypothesis, because all our measurements are very finite. But the whole of physics, amazing as it is, is nothing but a mathematical model. In the reality we measure there are no points, no axis, no vectors, no Hamiltoneans, no wave equations, etc., etc. These are part of these wonderful mathematical models that, for reasons no one really understands, provide us with very accurate predictions of how the world behaves. And it turns out that many of these very successful models use uncountable objects.

The number of potential states of geometric frustration for all potential bosonic and fermionic particle interactions within the Hubble sphere since t=0 would be considered uncountable because the limit of this sum diverges. I don't know if that could be construed to include the multiverse model because t=0(Big Bang) to t=today(2016 CE) is finite.