Searching hot designation text

Is there anyway to have the template search function be able to search what you input into the hot designation heading or text?

Search is joomla default search and iid functions as that but there are 3rd party replacement that could serve your need

Hi, is it possible to have the results appear on an empty page? Instead of with the modules and articles on the search page?

Since search page does not have its menu item it will use menu assignment as for home page. Advanced module manager you can install form extensions.joomla.org could give you ability to control that better.