each browser has find on page functionality (ctrl+F). Is there a way to detect user searches in javascript so that I could attach additional actions.
Of course you can try to hook into ctrl+f or cmd+f shortcut, but even if that works on "some" browsers, that way you only would know that an user pressed that shortcut and is most likely to search for something.
If the browser allows to overwrite that shortcut, you could further block the default behavior and implement your own search logic on the site. But that is considered most times as very bad practice. Overwritting native browser behavior is pretty pretty bad.
On the other hand, there is no "event" which gets triggered when a browser executed a search process. In short words, no, there actually is no way to detect or hook into a search process with javascript (if there is one, it's never gonna be cross browser compatible).