Driving in Taiwan (Taroko to Taichung)

Are there any restrictions on driving from Taroko to Taichung, over road 8 and 14甲 (or just road 8 through Baxianshan National Forest Recreation Area)? Google Maps doesn't even direct all the way through road 8, presumably due to the roadworks near Te-Chi reservoir (where the Google street view ends), but the road might be passable now?
Does one need a permit for driving through Taroko National Park, similar to the one, one needs to apply for when hiking?
Also, as this is a mountainous area, is a simple economy hire car enough for this journey or the gradients might be a bit too much? Just asking as when I was in Thailand the hire car I had was really struggling to make some of the hills.
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2021 Taiwan drive Taipei to Taichung|Driving music list|Formosa Freeway|09:00|這種天氣,開國道三號也有趣
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Images: michael spadoni, RODNAE Productions, RODNAE Productions, Timo Volz