You are right here. The Rematch string is part of the Info box, and as such it should be parsed correctly. (The string that contains the correct and breed/damage-specific min health requirements). This should be enough, but, I think there are also folks that aren’t using Rematch. They also should know of the type-specific min health requirements. Unfortunatly the default forms of the page don’t offer any detailed entries for the min health. They only offer one entry (for all damage types), and I always put the worst-case min health there (for pets that are weak against the expected damage type)
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Each battle that you win will yield experience for each pet that participated. However, deceased companions will not receive any experience (so make an effort to keep them alive by swaping in a different pet). Experience gains are based on your pet's level compared to the wild pet. Defeating a higher level opponent will grant more experience than a much lower level one, but be ready for a tough fight!