You'll need to create a Joomla menu link to CBs registration page then assign it as your homepage. You can't include the registration into a Joomla article or anything of the sort. CB has no registration module either.

Ajax page load vs a standard page load has no conversion difference; it just looks prettier. They're both still page loads and both have the same load times. We will of course being using much more jQuery in combination with Twitter Bootstrap for CB 2.0, but I can't say specifically how the workflow will behave at this time.

The current workflow seams perfectly fine depending on the gateway you use; for an on-site gateway it's about 3 clicks. For an off-site gateway it's upwards of 5 clicks and out of all the purchases I've done online (thousands) it's the same process everywhere (unless I'm already logged in of course).

You need to be reviewing the contents of your site and its outwards appearance to users as well as its ease of use and not blame the payment workflows for poor conversion rates; you can confirm this by using the new Piwik CBSubs plugin for Piwik (feee and open source) tracking that can track conversion rates and how many payment baskets are abandoned to see if the CBSubs process is in-fact what discourages users.

My immediate feedback on your site is it's over-bloated with content. For example the banner is huge, unnecessarily, and distracts from the content followed by an auto-play video (no one likes auto-play videos; it's as bad as playing music in the background). Menu links have far too much text and should be restricted to 1 descriptive word. Immediately I have to search your site for a registration link and then when I click "Get Access Now" expecting to get access I get an error page. When I finally do find a link that says "Become a Member" it again doesn't take me to a registration page; so that's twice now you've given me a false registration. At this point or even before it I'd close the site and never look back.

Note, please don't take my feedback as hostile and am simply trying to help you improve conversion rates; I'd be glad to provide more feedback with more time spent if you'd like. I also recommend contacting a UI expert that can give some advise on site layout.

Piwik looks interesting, but is it any different or more effective than Google Analytics at tracking conversion metrics and cart abandons?

Piwik is significantly faster at tracking then Google as it can be hosted on the same server as your website so you've instant analytics. We've also added a lot to its CBSubs integration for detailed eCommerce tracking (for example you can see when a basket was abandoned). It also tracks conversion rates of URLs and plans. So if I visit the plan page for Plan A, but don't purchase that's a failed conversion. Google is good, but I find it can take 24-48 hours sometimes to report analytics.

It really would be useful to have a CB registration module though... I may post a development job for this.

I agree and it's possible using CB API, but depending on how large your registration is this could be a really bad thing. For example if all your registration is is Username, Name, Email, Password then it'd be fine, but if you've a bunch of other fields then having it in a module would be a nightmare.