How can one concentrate on a koan whole day long while at the same time being mindfully aware of what is happening the sounds of the birds or bodily movements? I find this especially puzzling with regard to things that demand attention like cutting vegetables having conversations or teaching.

This is a simple answer. The meaning of concentrating on one´s koan is, to be touched by one´s own life energy. When we cut vegetables, when we cook, when we clean, this is easily possible. But when we talk to others, we easily lose track of our own center. Also when we think a lot about things, we cannot feel our life energy well.