Can I stream videos to Apple TV using the VLC iPad app? Can I stream videos to an Apple TV using the VLC iPad app?

In other terms, can I browse through movies on my Mac and stream them to Apple TV?
Unfortunately not. The VLC app was yanked from the app store some time ago due to some political issues within the developer base and conflicts with the app store policies. It hasn't been updated since then, and likely won't in the future. Also because it plays video in formats the iPad and Apple TV don't support, it's unlikely AirPlay support could be added anyway.

You might be able to use the iPad HDMI adapter to mirror the display on the TV, but I'm not sure how that performs with video going.

It's a bit outside the scope of AskDifferent, but there are several boxes similar in concept to the Apple TV that support a lot broader range of formats (much closer to VLC's level of support). I'm not aware of any that interface with an iPad, but most play from an attached hard drive or a network share. The WDTV Live and the Patriot Box Office are two popular ones, but there are many more.