I've driven Hwy 9 in a car. Very scenic. I guess if you were heading north then the sea would be on your right so it's easier to pull over! This road is covered by Google Maps Streetview, so you can just google up what it is like.
I don't know where you normally live but Taiwan is one of the few places in the world where it *is* a good idea to obtain an International Driving Permit from your home country. Don't know anything about bike rentals sorry. It seemed unusual for foreigners to rent cars in TW going by the reaction from the rental company, I'd guess renting bikes would be really really unusual.