Thanks for this question, it is always possible to further optimize Participants Database, and this is something I need to continue working on.
It isn’t practical, due to the way WordPress works, to avoid some operations on pages that don’t have a plugin shortcode. This is because at the early stages of the page load, the application has not yet loaded the page content, so you don’t know if there is a shortcode or not at that point. Waiting until you do know if there is a shortcode is too late for many things that need to get done on each page load. It’s fairly complicated.
The main suggestion I would make for your situation (on a host that has an apparently low limit to the number of db connections) is to use a page caching plugin. This will drastically reduce the number of db queries on most pages. You shouldn’t use page caching on specific pages with dynamic interactions, such as signup forms, but it will be easy to configure that.
Another thing to look at is if it is possible to reduce the number of fields you have defined in Participants Database, if you have unused fields, you should delete them as they contribute to the plugin overhead.
Meanwhile, I will look in to reducing the number of db queries, I have done this with the plugin many times before, but it is an ongoing process.