I certainly can’t answer all your questions, but here are a few thoughts. First, about driving: the only point would be to explore out-of-the-way places and make stops at your own pace. If the point for you is to get to particular places, take trains and buses.

If you’ve really just got one day for Taipei, I would suggest something like: morning to the National Palace Museum (you can have a supposedly Qing dynasty lunch there); afternoon to Taipei 101 (you can go to the top around dusk for day-night views); then Longshan Temple in the evening. There’s a night-market right around the temple, so you don’t have to go out to Shilin.

As for driving to Taroko, I would be tempted to reverse course (or take the coastal route both ways), so that you are driving up (northward) along the east coast rather than down it. That way, it's easier to stop at the little parking lots for sea views and walks, since they’ll be on your right and you don’t have to cut across traffic each time (assuming you may make a lot of stops). I guess you can get from Tianxiang (Tien-hsiang) to Taipei in 4 hours; Taipei-Yilan is well under an hour with the new highway, then it slows down. But if you drive up the coast with a few stops and walks, it could turn into 6 hours or more.

Jinguashi is lovely, but if you’ve been stopping along the ocean already, it’s really not necessary. Jiufen is very crowded. Very, very crowded, all the time. My favorite NE coastal stops include the Longdong-Bitou stretch of cliffs and the Miyuewan (Honeymoon Bay) beach, which is kind of funky.

For misty bamboo-cypress forests, don’t forget Alishan. You could take the high-speed train to Jiayi and then drive or train up to the mountain for a couple of nights. If you decided to visit Alishan, you could potentially make Taroko Gorge a one-day trip by flying TPE-Hualian. Or 2 nights in Alishan and 1 in Tianxiang (you could rent a car in Hualien just to use in the county). Jinguashi/Jiufen is also a one-day (or half-day) trip from Taipei.

Driving here is a bit tricky at first. You have to remember that the motorbikes whizzing around you in all directions (and from all directions) are counting on you to continue moving forward at a steady pace. In the mountains, the roads are *extremely* twisty and narrow and there are usually bits falling down the cliff, so don’t go too near the edge. Also, keep an eye out for rocks that have fallen on the road from above. Basically, driving is easy enough; just stay alert. Even without knowing Chinese, you should be OK with a good guidebook and maybe some google maps printed out with characters (to show people when you get lost). Most road signs have romanization of one sort or another.