Well I would suggest searching for each individual item…
I can tell you also that we (the community) probably won’t know where exactly to edit the HTML for the area under the search bar since the theme has been changed so much. (probably somewhere in app/design/*/*/catalogsearch)
MyCart and CompareProducts can be gotten rid of a lot of ways - do you only want them gone on the home page?
The home page is a CMS page, editable in the admin section (Admin -> Manage Page -> CMS Pages in the General Info tab, you can add and edit the text (specifically the “Home Page” text)