Unless the context is clear, I cannot promise you but,
1.birth order: 出産順位、年齢順
2.develop your talents: 才能を伸ばす
3.have an influence on someone: ~に影響を与える
4.introduce children to: 子供たちを~に紹介する
5.live up to his name: 期待に応(こた)える、彼の名に恥じない
6.the prevailing influence: 多大な影響
7.provide the environment: 環境に与える
8.realize your potential: 潜在能力を発揮する
9.treated as the baby of the family: 末っ子のように扱われる
10.child-rearing practice: 子育ての練習
If you have not tried 英辞郎 (Eijiro), it's worth visiting (online dictionary).