For the cork 'yes,' with one major exception: "I'm wondering if some "Elmer's Glue" and a bit of cork dust mixed together might bond & fill the cracks in the cork?"
Use Gorilla Glue or something that's really water proof. Remove cork from that 3/4 empty single malt Whisky bottle, pour into glass and consume while doing the following.
Take a fine file (even your sig-others nail file will do) and sand down the cork (remember, you don't need it any more) and add just enough 'glue' to get a simi-thick paste. Apply to breaks/holes in cork and let dry/set for several hours.
"Haste makes waste" here.
If it budges (probably) use 0000 steelwool
(or equivalent sand paper) to smooth out. Gentle here, a little goes a long way.
Which is why I recommend the 0000 rather than SP, far less aggressive.
As for the rest of the rod? Is it just 'cosmetic?' If so I'm not too sure I'd get all that involved save for a bit/drop of rod thread epoxy where needed.
As for Auntie Em's description above, I can assure you it works. Just a real time investment.