For the following document library can I delete the controls in the box?

Click on Edi Page and then Edit web part. You can rid of new document link by chaging ToolbarType to No Toolbar in List Views. The search box can be removed by going to Miscellaneous section and unchecking Display search box option. All Documents link will remain though. You may need to do some css tweaking for hiding that.
css tweaking required? 1.Add that Document Library as WebPart
2.Than Edit Web Part
3.In Properties List View, Select Toolbar Type As "No Toolbar"
To remove the both of the links for One library I would suggest : Open that page in firefox. Using firebug find out the class in which you can add css like Display:"none"
Edit page (AllItem.aspx)
Add WebPart (Media and Content) => (Content Editor) => Edit Source In this add Style tag copy css and done. Links will be not visible for that list.
E.g. In my case I found using firebug 2 classes and I added them in Content Editor web part.
<style type="text/css"> .ms-list-addnew-aligntop { display: none !important; padding-top: 0 !important; } .ms-csrlistview-controldiv { display: none !important; padding-bottom: 9px; } </style> This worked for me.