When you review first posts or possible edits, you can actually see progress toward review badges (and a few others). Is there a place that is similar but allows you to view progress toward all badges (where a progress measurement is possible anyway?) Should there be one?
No, not really.
The purpose with the obscurity is to keep us from gaming the system. The model of StackExchange is fun because humans are competitive by nature, it's what keeps us coming back to check our rep count every day. It's fun because it works, and it will only stay fun as long as it's fair.
Once people know exactly what they need to do to get badges, they will get busy with that. Sounds harmless, but it's not: deliberately or not, the quality of the site's content will suffer in the process.
As long as we don't know the exact measurement (or progress), we don't focus as much on getting the badge - we focus on getting rep, which we only get by posting good content. That's a trick of course, but it plays to our advantage: we get a site full of great content!
There are very few badges that are actively encouraged - these all relate to site quality; they are earned for editing existing posts.
I don't think so. If you search for "badge progress" on meta.stackoverflow.com you'll get quite a few hits, most of which seem to suggest that this kind of request won't happen. For example, see this question. However, as an answer to that same question points out, there are a lot of badge progress queries available at http://data.stackexchange.com although those aren't perfect. Also, I'm not sure if it works for beta sites.