Gussy, is a common golden gourami, 90% confident, as for the fins the were probably damaged when it was younger and were never treated properly so the fins never sealed so they would not encourage regrowth.
When we got George he was perfect but after a bit of establishing who was boss he had a chunk missing from his anal fin and lost 60% of his pelvic fins, we treated straight away for 5 days and the damage sealed quickly and the fins have now recovered and grown back, here is a pic with the damage taken a day after we got him. I'll post a new pic showing the healed fins later when I've sorted the other pc out, taking vista off and going back to XP so it's a bit of a job.
You could try treating it to see if would stimulate regrowth but I'm no expert maybe someone else can help out here, we treat at the first sign of any trouble, the earlier you can catch a problem the better chances of success.