To hide the search box, you'll need to edit your skin and remove the references to the [SEARCH] skin object.
To use a custom search tool instead of the DotNetNuke search tool, you'll need to put something in the skin that points to your search tool. The easiest approach might be to put a link to the page that contains the alternative search tool. Alternatively, you could take the code for the [SEARCH] skin object and create a [SEARCH1] that invokes your search tool. I've never looked at doing that, so I don't know how difficult that task might be.
Before you start anything, though, I suggest that you need to learn a bit about DotNetNuke skins, how they work, and how they are made. To do so, take a look at some of the
tutorials on skinning that are available here, including the
DotNetNuke Skinning Toolkit.