What benefits would there be in making the Villain a mirror of the Hero? Could the Theme be weakened because I would have to avoid my Hero choosing to change (or be changed) because the Villain is acting the same way; or, could I use the Villain to show the logical end of his actions?

Absolutely, yes--in many stories, the Villain is a mirror of the Hero--or, more accurately, a mirror of the Theme.

For stories written well around a theme (which many stories are, whether or not it's intentional), EVERY single character in the entire story reflects the theme in some way, down to the throwaway characters in the background. So in a well-structured theme-based story, the Villain will reflect the opposite of the theme (antithesis). This is not only for story structure and integrity, but also, as you noted, to further the Hero's journey.

It's not always an overt mirror--sometimes it's a very subtle relation to the theme--but it's there. But if you're stuck, or you feel like your villain isn't quite right, standing back and looking at your villain from the perspective of the theme can help a lot.

A good overt example is the movie WALL-E. If you look at the movie as revolving around the theme of "directive" (protocol, doing what's expected of your "programming"), you can see that Wall-E, Eve, AUTO, and even the gatekeeper bot whose job it is to run the elevator all struggle with following programming to varying degrees. It's a great film to study if you want an easy-to-spot, overt example of consistent theme.

Ah, that's the question! Personally, I really like it when hero and villain are mirrors of each other, because it forces the hero to see what they could become. The original Star Wars movies (especially the original trilogy) have this dynamic between Luke and Vader. For Luke to defeat Vader, he must become a Jedi, but becoming a Jedi increases the risk of him becoming Vader. We see this especially in Empire Strikes Back when Luke loses his hand. That comes back later in Return of the Jedi in the final duel after Luke severs Vader's hand. In his quest to save his friends, using his emotions (especially anger and hate), Luke took the first step to becoming Vader (eg. wearing all black and having a mechanical hand). Then he had to step back and decide if he was on the right path.

I have this in my own work with the superhero Liberator and her nemesis Justice. Both have nearly identical backstories, both have lost many of the same things, and both want the same thing. The difference is how they approach it.

You could do the same thing, because it constantly forces the hero to assess their decisions and actions. Are they doing the right thing? Or are they becoming the villain? It sets the story up for some excellent conflict when you do that.

I'm working this question in my story now. The main character wants what the antagonist has. She doesn't realize it that way, but in effect. Part of the story is showing the good character turning away from the temptations and growing as a person instead.