On a default style, links are modified in Admincp > Settings > Navigation Manager > edit link . You have a custom style and it could be that those links are just hard-coded. You can try getting rid of them in the Navigation Manager, but it that doesn't work, you'll have to find the template where they are and modify it that way.
The search bar is something you would need to modify via a template also (or you may be able to set display: none for both .globalsearch and .advanced_search).
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