Simply put does there have to be a cube next to the lower order chosen? If there is a cube present is it also removed?
No and no.
Just follow the procedure as listed under Rule 30 in the rulebook. Notice they don't mention any requirement for the lower order to have a cube or removing a second cube from the lower order.
You remove one cube. Pay the initiative as indicated in the box where the cube was removed. You now have a variety of orders available to you for this turn. The order where the cube was removed or any one of the orders listed below it. Select one order and perform it.