In the general case, the comparability of cardinals relies on the axiom of choice. In other words, they are comparable, but they require a slightly unintuitive foundation to establish that they are always comparable.
Not if you aim to pass your exam.
Sure they are. You can define a one-to-one mapping, they're equal.
You can define a one-to-one mapping between the sets {1 2} and {3 4}, but I don't think anyone would say they are equal.
You’re thinking of isomorphic, not equal.
They meant "their cardinalities are equal". It's honestly an easy mistake to make, especially if typing on a small screen. Or especially if having a discussion where sizes of infinity are already being discussed.