Yes, you can customize and order the columns on the Leads search page. The bulk action checkbox and the additional action columns however will always be there.
As an Admin user, navigate to the Admin > Setup > Application Configuration > More (tab).
Click on the Leads Search Columns link.
Select the columns you want to add from the dropdown and click 'Add'. You can re-order the columns by dragging and dropping the hamburger icon on the left. You can also remove columns by clicking Remove on the right hand side.