If a website is developed in a way such that no matter what the user does, the URL never changes and the page is not reloaded, how can we track the time spent on the page? If you wanted to compare the number of sessions and conversion rate per device type of a website to the one of other websites, is it possible? Thanks..
You could try adding Google Analytics to your site. It will give you most of the info you are asking for.
Rewrite the page. Having ajax-calls, or just a .htaccess that makes no urls show in the url-bar is just poor design. Adding javascript functionality to actually facilitate url-changes when doing something on the page isn't that hard, nor is changing the .htaccess, if that's the problem. It will also improve the user-experience, since they'll be able to bookmark subpages, instead of just the main domain.
I'm not sure if it is the answer but have a look at Google Analytics Virtual Page Views... basically you can change your analytics code in each file so that GA sees it as a page, even though it has the same URL. It's mostly used for event tracking e.g. click or download but might also work for you. There is different code depending on if you use GA or the newer Universal Analytics code. But as per the above post, I can't see that having only one URL is the best way to run your site as surely you are missing out on free traffic from SEO? The only instance where it could be a necessary evil would be the checkout, which my solution above is commonly used for. If it is a checkout you're looking to analyse, then a tool like Mousestats Form Views (and Playbacks) would also be useful assuming it has no issues with your set-up.