お父さんに愛されていると信じて疑わないように、周りの人も優しいのが普通だと思ってた
Yeah, ように mean "as/like". So this must translating like "I thought that it's normal when people around are kind to me like believe that dad loves me." ? Or not?
Yes, ように means "as/like" here, among a few possibilities. This is a minor issue, but I would interpret お父さんに愛されていると信じて疑わない is what happened in the past: so the sentence translates like "... like I believed that Dad loved me."
