After completion of the Companions quest line...is the containers in Kodiak's room (that you now have full possession of) safe to store stuff in??

i have not done that quest but from what i have read it is. just like the college and thieves guild flaggon. just try it with something you dont care to lose. leave it for a couple saves. if it stays try a few more things you wouldnt mind losing.

I have left books on the bed, and they are still there.

Yes things dropped usually stays but the question is if the inventory in the containers is re spawning,.