Beautiful tree!
I don't know about the variety or the flower browning, but yes, you can trim back branches if they're too long and slender. I would definitely wait until flowering has stopped on those branches, but that's just because I'm selfish and want to see the blooms as long as I can. If you live in an area with an extended growing season then you can still crop back this year, but it's usually best done in early spring when fresh growth has begun. Once again, beautiful plant