Can I use the chests in Harmondale to store question items, or other stuff, or will the stuff disapear after awhile?

AFAIR you can safely do so. I guess that's the purpose of those chests. Don't worry about quest items though - if you ever loose is just visit the arbiter and he will give it back to you.

I can confirm that my stuff in Harmondale Castle chests was there even after several months of in-game time.

Castle Harmondale is the only place in the game that won't respawn, so go ahead and store whatever you like there :)