Is it possible to view our personal stats on the site (number of x until level y or badge z)?
If so, how?
If not, is there any feature being worked on for this?
For privileges, go to https://judaism.stackexchange.com/help/privileges?tab=all.
Badges are described briefly at https://judaism.stackexchange.com/help/badges?tab=general&filter=all and more fully at http://meta.stackoverflow.com/q/67397, but I know of no way for a user to know how close he is to all the badges.
There is info on some badges, though:
The Reviewer badge requires one to issue 250 reviews of the same type, and the Steward badge requires 1000; you can see how many (for example) closure reviews you've issued at https://judaism.stackexchange.com/review/close/stats.)
You can check your status regarding the Civic Duty, Copy Editor, Custodian, Electorate, Reviewer, and Steward badges by hovering over the percentage-done bar to the right of the "Review" header on any of the review pages (e.g.).
You can check your status regarding the Tenacious and Unsung Hero badges on SEDE here.
You can check your status regarding the Pundit badge on SEDE here.
You can check your status regarding the Civic Duty, Electorate, Deputy, and Marshal badges on your user page.
You can check your status regarding the Epic and Legendary badges here at the bottom.
Stinking Badges! is an unofficial (AFAIK not SE-supported) way to see your progress toward certain badges.
This answer is community-wiki: it's easy for others to edit. Please add info on more badges to this list!