The analytics integration is available in the free version as well. I just activated the pro version search bar like 10 minutes ago on your site, so it had no effect on the analitics yet. I see you have the universal analytics tracker code available, so there should be no issues, your configuration looks okay too.
1. If you have any ad-blockers, or any similar plugins in your browser, it won't work when testing. I usually test in incognito mode, looking at the live traffic to see the search phrases within a 10 second delay.
If you you are not looking at live traffic chart, google analytics needs usually 4 hours to propagate the data. So searches done within the last 4 hours are usually not visible yet.
Make sure to clear all layers of cache you have. Usually what I do is I open a live analytics traffic window, then reload the site and check if that is registered as a page view. If yes, then I try a few phrases, if not, then it means that there is either an issue with the analytics tracking code, or an ad or traffic blocker is active.
2. Regarding the analytics, the code is exactly the same for both versions, so there is no difference in this feature, at least not for now. The upcoming release is focused on autocomplete, pre-cache and analytics event tracking integration. I'm working on possibilities to track every detail of how the user interacts with the plugin.Best,
If you like my products, don't forget to rate them on codecanyon :)