If I were to just take a hike in the forest in Taiwan, somewhere where there is no trail or any human presence at all, what would be the biggest threats to my health and safety I'd need to bear in mind? How should I prepare for them?
Frustration. Take some that wonder drug with you.
Seriously, if you're thinking about bushwhacking here, I'd say forget it. Number 1, you'll cover so little distance it won't be worth your while -- the forests are seriously overgrown with brush and you'll not get far, even with a petrol-driven strimmer, saw and machete. Number 2, in the unlikely event that you do cover any distance, it would be extremely easy to get lost among the many detours you'd have to take to avoid ravines, ridges, etc.
And there are beasties out there, some of which can bite and sting.
Be sensible and stick to the many, many trails.