1. It's not completely unnatural, but it'd be better phrased like this:
In addition to looking extremely good, she has a great character and I suppose she will make for a good wife for him.
Besides didn't really match the intended meaning thus the change. Also subject in english sounds monst natural when it's at the beginning! ( That's why the "I suppose" change)
2. Besided mistakes from above, there should be "she looks". When talking about someone's appearance, present continuous isn't used- She LOOKS extremely good.