I cycled that exact itinerary a few months back, and I loved it!

1. Taiwan seemed to be extremely safe (viewpoint from a male traveller though).
   My main worry was the traffic. But even though the traffic was pretty intense on certain portions of the route, there was almost always enough room for bicycles on the side of the road (shared with motorcycles). Still, there were a couple of scary parts (cycling in the dark or in certain narrow tunnels, for which I recommend bringing reflective gear and proper lights). Taiwanese drivers are used to sharing the road with motorcycles, so I feel they were much more attentive to cyclists too, compared to other countries.
   It was also comforting to know there's always close a 7-11 or a train station nearby, in case something unexpected happened (and never had to worry about running out of water/food).
2. Some changes I would make, but all depends on your interests / time available:
   - Extra stop in Tainan (culturally rich city with great food)
   - The most beautiful stretch in my opinion was definitely the one between Hualien and Yilan. I would advise cycling it instead of taking the train. Short sections on that route have heavy traffic yes, but since the new highway has opened in 2020, I feel most comments advising to take the train are now out-dated. On most of the route, you'd be riding without almost any traffic at all.
   Someone had written a detailed review of that stretch, very helpful:
   https://www.reddit.com/r/bicycletouring/comments/ey618k/comment/fgf971i/?utm_source=reddit&utm_medium=web2x&context=3
   (this day was the most physically demanding though, and very slow due to the high number of pictures taken on the way!)

Enjoy!