Firstly, kudo's for being interested in the cardinal numbers. However, there's a few errors in the question, so lets just try to set the record straight.
The (currently standard) collection of set-theoretic principles is called ZFC. So rather than asking "what is true of the cardinal numbers?" (vague question), let us ask "what can ZFC prove about the cardinal numbers?" (precise question).
ZFC proves the following sentences.
- $\beth_0 = |\mathbb{N}|$
- $\beth_1 = |\mathbb{R}|$ (but we cannot prove $\aleph_1 = |\mathbb{R}|$).
- $\beth_0 < \beth_1$.
- There exist infinitely-many infinite cardinal numbers.
- There does not exist a set of all cardinal numbers.
See also, beth numbers, aleph numbers.
To see why there ought to exist infinitely-many infinite cardinal numbers:
Step 1. Recall Cantor's theorem: $|X| < |\mathcal{P}(X)|$ for all sets $X$.
Step 2. Consider the following sequence. $$|\mathbb{N}|, |\mathcal{P}(\mathbb{N})|, |\mathcal{P}(\mathcal{P}(\mathbb{N}))|,...$$
(Of course, this is just the initial portion of the beth sequence: $\beth_0, \beth_1,\beth_2,\ldots$)
I might have a go answering some of your other questions a bit later.