Have reviewed your install. First issue is your top menu login, register, and logout menu items were to Joomla and not CB. I've corrected this for you. Next issue is you're using Joomlas page cache. Page cache doesn't work in dynamic environments or on pages with HTML forms due to their single use security tokens. I've configure it to exclude CBs profile edit, login, logout, registration, and forgot login pages so it won't cache those anymore; you may have issues elsewhere with this so this is a heads up. After clearing cache registration appears working perfectly fine.
I also noticed you've something installed injecting into the HTML of ajax requests. I've no idea what's doing it, but you'll see it happen when the username field tries to validate (you'll see the raw JSON). I did not fix this for you. You'll need to investigate that your self.