Melding pets allows you to have a pet with better stats. So if you wanted your target pet to have higher HP growth you'd get another pet which will be the source with higher HP growth. And it'll allow you to slightly increase overall stats. Different attitudes increase different stats. If done right you can end up with some pretty godly pets.
Mutations I have no idea how it works... there's someone from the Chinese version who's sharing the info on officials. Her name is petprofessor... xD It is possible to get a pet that isn't tamable in the wild through pet melding like... a haunten type for example.