AF(phi1 & Xphi2) is not a CTL formula (regardless whether valid or not), since CTL formulas are defined as those CTL* formulas where a temporal operator occurs right after a path quantifier and where after each path quantifier there is a temporal operator. There is however no path quantifier in front of the X operator since there is a conjunction, so it is not a CTL formula. Even worse: AF(phi1 & Xphi2) is not equivalent to any CTL formula!

AF(phi1 & AX phi2) is a CTL formula provided that phi1 and phi2 are CTL formulas.

However, X=AX is not correct, please remove this from your cheat sheet.

The point is that AF(phi1 & AX phi2) and AF(phi1 & X phi2) are not equivalent. The formulas look very similar, but are not the same. If you continue reading the slides, you will find on page 38 a Kripke structure that satisfies one of the formulas, but not the other one.

Fa & b = b & Fa is certainly valid since conjunction is commutative, and since disjunction is also commutative, also Fa | b = b | Fa is valid.

However, Fa | b = Fa is not valid, and neither is Fa & b = b. You can find easily counterexamples with the LTL teaching tool.