The Pauli exclusion Principle states that two Fermions of the same type cannot exist in the same state at the same time.
And place. It's to do with atomic orbitals. However you sometimes hear people saying it applies to fermions throughout the universe. That's a popscience myth. See this question
This means that two electrons cannot both exist in the same spin state and be in the same location at the same time.
Good stuff. I think it helps to think of spinors, and remember that two ocean waves can ride over one another, but two whirlpools cannot overlap.
This is also the reason that solids tend to be unable to pass through each other.
I'd say it's part of the reason. Co-rotating vortices attract, counter-rotating vortices repel. And electrons repel one another.
I was wondering if there were two pieces of frozen hydrogen in which all the electrons of one piece were spin up and all the electrons of the other were spin down would these two pieces of frozen hydrogen be able to pass through each other considering that all the electrons in one would have an opposite spin from all the electrons from the other.
They wouldn't. You can see a depiction of spin up and spin down on Rod Nave's hyperphysics website:
Imagine you had two hydrogen atoms next to each, one spin up, one spin down. Their electrons will still repel one another. But see Gert's comment. They can also make a molecule.