@valtsai - Assuming your "search" event fires on load of the results page, and the page loads enough to fire the call to Adobe, then you will see all metrics and dimensions collect and increment in reports. If the page is reloading before the call to Adobe can fire, then you will not see the data collected. You can test this by checking in your preferred debugger or the network tab of your browser's dev tools.
Regarding tracking page load time, I'll point you to the getPageLoadTime() plugin: https://experienceleague.adobe.com/docs/analytics/implementation/vars/plugins/getpageloadtime.html?l...
To track the page load issues/scenarios you mentioned, you'll won't be able to use Analytics. You'll have to find a different service to monitor for issues. For example, if the page doesn't load at all, your Analytics script will not run and no call to Adobe can be made.