> does your kidney compete with your lung?

I don't know of any cases of direct kidney v lung competition, but competition among body parts is common. Sometime that competition is adaptive, sometimes not. Examples of adaptive competition are things like when under extreme circumstances (particularly cold or hunger) your body will sacrifice parts of itself to keep other parts going. Examples of non-adaptive competition are things like autoimmune diseases. Also, sometimes individual cells go rogue and stop cooperating. That's called cancer.

> competition only seems to be important at the very edges of what selection is acting upon

I guess that depends on what you consider "the edges". In the case of humans, selection produces intuitions about "us vs them". Those intuitions range from very closely drawn boundaries ("us" includes only my immediate family or clan) to very broadly drawn boundaries ("us" is my entire species, or my entire phylum, or all living things). In between are things like "us" is all members of my species with my skin color. But the extremes of this range are non-adaptive. Draw the boundaries too narrowly and you end up without enough genetic diversity in your in-group to drive out maladaptive mutations. Draw them too broadly and you end up defenseless against parasites and with nothing to eat.