Asaf certainly covered the formal parts of it; I'll try to answer what I believe is the underlying issue in understanding.
For real numbers, we have a rather "intuitive" way of thinking what "all" of them are - every possible combination of digits in decimal, for instance.
But when we do formal mathematics and set theory - which we need to if we want to talk about concepts such as infinite cardinalities, then things are more complicated than that. What does the above mean when formalized?
There's a simple fact to be considered here: Our language - natural language, formal language of logic and pretty much anything else we can actually work with - is finite in nature. This means that no statement that we can make can distinguish every single element of an uncountable set. It's not a matter of the "right" language - whatever we humans can do and understand is finite in nature, fundamentally.
This also applies to set theory, and so what happens is that certain models of set theory leave some (or rather most) of these intuitive "real numbers" out simply because there's no formula to describe them (or more precisely, distinguish them from all other "real numbers"). And the same is, of course, true for bijections between (infinite) sets.
And since cardinalities basically are all about the existance of bijections (with the natural numbers, the reals or whatever else), the model of set theory we choose can change which sets and bijections "exist" and thus the answers to your questions.
What does that mean for the "real reals"? That, in the end, is a matter of belief.
Many mathematicians will take the pragmatic approach and work with what they need.
It may seem worrying that this might mean their "reals" miss numbers - but consider that they are only missed because there is no way whatsoever to describe them. No formula, no definable function, not a zero of those or anything else. The fact that we have no way of describing them means they cannot cause any problems when missing, too - because if they did, we'd have a way to describe what is missing and thus the number in question.
Of course, there also is the realist's view that the "reals" as intuitively described exist, and in fact, many mathematicians will likely take that point of view unless they need set theoretic formalism for something.