How should I protect wooden trim that abuts a brick chimney from further water damage?
Well, I'll answer my own question. So I scrapped all the loose paint away, and used 4 inch deck screws (barely counter-sinked) to more firmly attach the crown-molding back to the house. I filled in over the deck screws with wood filler to hide them. I caulked the gap underneath the molding. I picked out all of the rotten wood at the end of the fascia board, where it abuts the chimney. I applied Minwax Wood Hardener to the end of the fascia board. I then carefully folded up a section of zinc-coated steel plumbers strap (AKA "hanger bar") about four times to match the shape of the missing wood [it looked like a reversed "C"]. Folding the strap multiple times insured that the gap, between the sections of strap, was small. I pushed the tape in so that it was just below flush with the fascia board. Then I used Minwax Wood filler and applied 4 coats (eventually) to fill in the end of the fascia board, using the plumbers strap as a substrate for holding the wood filler. Then I primed and painted. 

From the ground it looks good--you'd have to really stare at it to know that something was repaired there. I left a little gap between the end of the fascia board (now consisting entirely of wood filler and plumbers strap) for water to drain down. I didn't do any caulking of the siding. I also extended and angled the shiny metal flashing (visible in the photo) so that water could not easily flow/drip behind the molding. And in case your wondering, yes, that chimney needs to be repointed. Add it to the list...