Is the language that accepts strings concatenated with their reverse regular?
You're over-thinking it on the $xw^R=ww^R$ thing, that's a red herring. But you're not over-thinking it by looking for a contradiction, you're looking in the wrong place. 

First, your proof that the language is irregular is correct. You're aiming to show a contradiction. All you need is one counterexample; just one! And you found it already. This may seem paradoxical, but your understanding of how to operate on languages is a little off. As you noted, regular languages are closed under reversal:

$L^R = \{w^R|w\in L\}$

They are also closed under concatenation:

$L\circ L^R = \{wv|w\in L, v\in L^R\}$

But, you see, that's a little different from $L_{mirrored} = \{ww^R|w\in L\}$. Look:

$L = \{$ cat, dog$\}$

$L^R = \{$ tac, god$\}$

$L\circ L^R = \{$ cattac, catgod, dogtac, doggod$\}$

$L_{mirrored} = \{$ cattac, doggod$\}$

There's your contradiction! You must have thought those last two were the same, when they're not. Rather,

$L_{mirrored} \subseteq L\circ L^R$

And the subset of a regular language is not necessarily regular.

edit: I removed a large chunk of this answer, so the comments below will probably be confusing...