1)
It's a bit classical imperative of する, i.e., しなさい (a polite prefix お + the -masu stem of する). Old people, especially women often use this form.
2)
Yes, 根性悪[こんじょうわる]の and 老いた both modify ブルドッグ. It's the same meaning as 根性の悪い, of course.
3) 4)
Right.
5)
が indicates the object (what the subject envies), but, confusingly, can also indicate the subject especially in a subordinate clause (e.g. 私が羨ましいのは彼の能力だ). It's the same as the adjectives such like 好きだ, 憎い or 面白い.