I believe that in order for this question to be clear, the OP should provide the exact definition of "hyperreals" that is being used in the question.

Nothing in the question hints at any object that has "faces".

You make no mention of x other than to state that it is a member of 𝛀. It would help me understand the question if you clearly distinguished between what is the unknown that you are trying to solve for, and what is given.

Yes, always read actively. With a pen or pencil and a pad of paper, so you can try to check anything in the book that seems less than obvious. Also, look ahead and at the table of contents, so you have an idea of where you've been and where you are going.

It's much better to let the OP answer such a question, especially since the linked article has multiple definitions for "almost disjoint".

Have people considered adding "𝜅 < 𝜆 ⇒ 2^𝜅 < 2^𝜆 as a axiom extending ZFC ?

Do you mean "minimum" where you write "max value" ?

Can we assume that the word "nonoverlapping" here means precisely that the interiors have empty intersection?

I have not read previous comments, but: The question of proving that "N^x is an integer for all positive integers N" implies "x is an integer" was on an old Putnam exam, probably in the early '70s. The suggested solution made use of only this fact for N ∈ {2,3}, just as does the purported unsolved Ramanujan problem. I don't know if Ramanujan thought about this problem or not, but it definitely has been solved. Let me add, one the hardest Putnam problems anyone could remember. I don't recall whether anyone solved it during the exam.

Namely, Hopf tori in S^3, Ulrich Pinkall, Inventiones Mathematicae, v. 81, 1985.

Ian — I had thought that result applied to conformally equivalent Riemann surfaces (rather than isometry classes up to scaling).

When you mention "the flat torus", which of the infinitely many conformally inequivalent flat tori does this refer to?

There is no such thing as the flat torus. There are uncountably many distinct flat tori that are not mutually conformally equivalent. (The flat torus that has been C^1 isometrically embedded in 3-space explicitly is a square torus.)