If I enter Taiwan only after my Gold Card is approved, will it affect the start date for applying for permanent residency?

No. The start date for permanent residency eligibility under the Gold Card is based on the issue date printed on the card, not the date you collect the card. As long as your Gold Card remains valid for three years, you reside in Taiwan continuously for three years with an average of at least 183 days per year, and you meet other application requirements, you can apply for permanent residency.