Thanks, Munroe, this is what I found:
It really doesn't give much more than skeletal information: no history, famous founders, inspirational legends, and such. Anyone with more info please direct me to the source!
So, to question one: such an organization does exist.
Question two: No racial restrictions, but Paladin-only class restriction.
Question three: Membership being limited to paladins, there is still the question of association. How much would be proper without DM intervention? Since the organization does not exist in FRC, does it matter?
Question four answered too.
By the nature of Kelemvor and his worshipers I'd expect his church to be rigidly controlled by a central bureaucracy with each temple, shrine, and monastery linked. The Knights would be an elite organization within the church.
I assume the knighthood, being new, is rather thinly spread, perhaps with a central fortification near a former undead threat serving as home base. More commonly, individual knights, served by squires and pages, with Men-at-Arms if needed, occupy some temple near an undead threat.
P.S. I seriously want that power that cuts through all but epic DR!