Problems in understanding the solution for "Mermin–Peres Magic Square Game"

I am referring to the article Advanced Topics in Quantum Information Theory exercise 4 and to the MS Quantum Kata MS-Quantum-Kata which describes a solution for the Mermin–Peres Magic Square Game.
I could follow the article and do all the maths for myself until the end of the article.
Unfortunately the article ends with

… Following the hint, we consider two maximally entangled states shared between Alice and Bob:$$|Ψ〉_{AB}=1/√2(|00〉_{A_1B_1}+|11〉_{A_1B_1})⊗1/√2(|00〉_{A_2B_2}+|11〉_{A_2B_2})$$
It is then easy to check by using $$1/√2(|00〉+|11〉) =1/√2(|phi^x_0phi^x_0〉+|phi^x_1phi^x_1〉) =1/√2(|phi^y_0phi^y_1〉+|phi^y_1phi^y_0〉)$$ where $|0,1〉$ are the eigenvectors of $σ_z$ and where $|φ^{x,y}_j〉$ denote the eigenvectors of $σ_x$ and $σ_y$ as in the solution to the last exercise (with eigenvalues ±1 each), that the compatibility condition is indeed satisfied.

Unfortunately for me it is not easy to check, I spent multiple nights to try this, but I don’t get it done.

1. Can anyone show me how this check looks like? What is the math behind it?
2. The algorithm states that Alice (and Bob) are performing multiple measurements on their qubits for all observables $M_{x,y}$ in her row (in his column).
   I don’t understand why this makes sense: After the first measurement the quantum state has collapsed (or could have at least), why does it make sense to do additional measurements?
   I know there is something called "phase kick-back" that kicks the phase into the control bit if I use "Controlled Operations", is that something that is used here? How exactly?
3. Is "Phase Kickback" only a mathematical trick or do physical systems really behave like this? Is there an easy to understand sample maybe, for a non-physic, non-mathematic guy (just a software dev)?

Thanks a lot for any help or hints.