"Find an Item" box: To hide this, go up to the Gear icon at the top of the page, and choose "Edit Page" to put the page into edit mode. Once in edit mode, the upper-right-hand corner of the List will have a little drop-down arrow. When you click that arrow, you can choose the option to "Edit Web Part".
This will open the web part properties pane on the right-hand side of the page. From there List views go to the Miscellaneous section. From there Uncheck the checkbox "Display search box". Then click the "OK" button at the bottom of the properties pane.
Top Links: There is no built-in setting to "hide" or "turn off" this set of links, but if you hit the "Edit Link" pencil icon next to those links, you have the ability to Delete each one manually.