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I will post an answer to my own question, based on the information I received elsewhere.
The information about current condition of the mountain highways can be found on the Taroko National Park website.
There should be no major road works on Highway 8 now, given that Bus 6506 is running the entire highway from Taichung to Lishan.
No permits are required to drive through Taroko National Park.
Most sedan cars should be fine but one must be comfortable driving in snow/ice and using snow chains in case it has frozen over in winter months. Manual transmissions recommended in steeper sections especially between Xinbaiyang and Wuling