OK, if Tenchi-ken is Funaho-ki's key and Juraians need keys to interface with their
tree, why give the sword to Tenchi?

He is in effect, giving up his control over Funaho.  This also seems puzzling since
we mostly seem to agree that Yosho didn't give Funaho-ki to Tenchi.  The reason most
folks think that Funaho-ki was given to Tenchi was because Tenchi-ken was given to

him.  If Tenchi-ken was Funaho-ki's, one can understand why they'd think that. 

While I can't account for 100% of keys used, all of the ones we have seen used have
been in physical contact with the user. Tenchi did use Tenchi-ken to activate Ryu-oh's
LHW's, but he had to be inside Ryu-oh's container to control her AND he was all
but touching Tenchi-ken. 

Second, why did Kagato recognize Tenchi-ken?  Why did he call the sword the "king's
property" (OK, I'm being forced to rely on Pioneer's translations but that's all I
have to go on at present)?  And how did Kagato get the idea that Tenchi-ken was what

he wanted?  If the sword was Funaho-ki's, it wasn't around all that long before disappearing.
And did Yosho do anything with it (battle wise) prior to his battle with Ryoko?

Oh, TS -- do you have access to read question 13 from "101" and do a re-translation?
You quoted the old fan translation and I would like you to clarify it.

"Therefore, Youshou's key (Tenchi Sword) is a master key while Aeka's is just a normal
key. This is why Tenchi, with his master key, could release Ryoko from Aeka's jail
in ep.2."

Doesn't this sound like two master keys in question?  I want to see if the translator
added the "Tenchi Sword" into the translation.

:-)