I just noticed your addition to the first post - it's a huge improvement. The whites are peaking too high (too bright, in other words!) There are two areas where white seems to be at 100% and all detail is lost. One place is the reflection on the thingy hanging off the collar. That's OK because it's reflecting a light source. The other is the very light areas of the fur around the head - they aren't light sources so you can safely reduce their brightness value to 92-95% and regain some detail there too.
In general, the lightest area in a photograph needn't be 100% white. Obviously if you have a photograph of a landscape on a cloudy afternoon, it's unlikely that anything will be very bright. But if there's a car in the pic with its headlights on, you can safely adjust the brightness level so the headlights just reach 100%.