As I read one of your entries, certainly there are Senior Rabbis of congregations that are Kohainim. How do they officiate at the funerals of their congregants? Their presence is essential to the living and the rabbi of a congregation has a responsibility to his congregants.

Can you please cite examples?

In the Orthodox way, a rabbi of Kohanic lineage may officiate at a funeral. However,he may not be in the same building with the dead. It is customary, when a Kohain officiates, that the deceased remain outside the building (i.e. in the hearse). At a cemetery, the rabbi may officiate from off the grounds, while the deceased is at the grave site.
Attending to the living has no restrictions at all. The rabbi may console the survivor and conduct Shiva services at the home of the deceased.