Where in the Chomsky Hierarchy are Regular Expressions as a language?
Are we talking regular expressions as in only union, concatenation and star?

Consider the following grammar:

$R -&gt; a | b | c$

$R -&gt; R+R$

$R -&gt; RR$

$R -&gt; R^*$

$R -&gt; (R)$

This captures all regular expressions. So if we're only talking in the Chomsky hierarchy, then it's clearly context free.
If you allow parentheses, it's not regular, since well-nested parentheses are known to not be regular. If you want a proof, do a homomorphism from the language of REs to the language of well-nested parentheses.

I don't know where they fit in terms of more detailed classification (i.e. $LL(1)$, $LR(0)$ etc.).

You could probably find something about this if you searched for "parsing regular expressions."