Differences between $\omega$ versus $\aleph_0$ In what sense is $\omega$ different from $\aleph_0$?  Is the following set of statements true?
(1) $|\omega| = |\aleph_0|$
(2) $\omega$ is imbued with an ordering, while $\aleph_0$ isn't (although can be). 
(3) Both are (can be?) constructed from the empty set in the canonical way. Hence their underlying sets are equal to each other.
 A: When formalised within ZF(C), with the usual (von Neumann) definition of ordinals and cardinals, they're the same in the sense that they contain the same elements. But I like to think of them as different things:


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*$\aleph_0$ is a cardinal number; it refers to the size of a set.

*$\omega$ is an ordinal number; it describes the order relation on a set.


The arithmetic we do with cardinal numbers and ordinal numbers is different, for instance


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*The successor of $\aleph_0$ is $\aleph_1$, which is the smallest uncountable cardinal.

*The successor of $\omega$ is $\omega+1$, which is the order-type of the set of natural numbers (under their usual ordering) with a point added at the very end.


From a notational point of view, using $\aleph_0$ vs. $\omega$ is probably most important when you're doing cardinal and ordinal arithmetic, but using them both properly helps keep track of the meaning behind what you're saying.
As regards your questions: all the alephs (i.e. cardinals of the form $\aleph_{\alpha}$ for ordinals $\alpha$) are well-orderable cardinals, precisely because $\aleph_{\alpha}$ is (by definition!) the cardinality of $\omega_{\alpha}$. Thus even if you don't take the von Neumann definition of ordinals and $\omega_{\alpha}$ and $\aleph_{\alpha}$ happen to have different underlying sets, if we take $\aleph_{\alpha}$ to be a set then it will have a natural well-ordering of length $\omega_{\alpha}$.
