Sorry the dumb question, but why not?
Can't I make a static page that offers search (reload on every query with results) or profile (reload every time soemthing is changed in the profile)
A static site does not have any backend, thus cannot provide you with search results because it cannot do any computing at all (just provide you the same (static) content again and again).
You can go around these by having some presets in the profile and providing cached results for the most common search types but this is more or less as far as you can go.
I can change the values of the presets I can add headers, remove headers change the value of cookies, add cookies, remove cookies etc etc.
Most people don't need overly complex searches...