Hello! Congrats on your new family member. I adopted my first amazon this past weekend... and they are amazing! (In fact, my new friend Henry is sitting on my shoulder right now as I type!) "Rio" is so lucky to have found you to adopt him... someone who is already invested in his well-being from day one!
I think renaming your bird would be fine. I have heard that people who rescue dogs will rename them because the dogs associate their old name with their crummy old owners. Maybe the same logic would work with a parrot? Regardless, parrots are smart and will learn a new name. On the other hand, I didn't rename Henry because she yells "hi Henry!" and I like the name.
I'd say sexing the bird doesn't matter too much. When I got my conure 16 years ago, I picked a gender neutral name with the intention of getting her sexed, but I never did (maybe because I was only 12!) and Jade has always been a "she."
Does "Rio" wander when you leave the cage door open? I've found that I don't need to lock Henry in yet. She prefers to be on top of her cage unless she goes in to eat, and since she doesn't try to fly down to the floor, I've just been letting her choose where she wants to be. Then again Henry is a Mexican red headed amazon, and they're known to be pretty docile compared to other amazons, so I don't know if a YC would be OK out unsupervised. Also depends on the individual too.
Did Rio's old owners ever handle him? He probably was handled by his breeders when he was a baby, which wasn't too long ago, so if he is scared of hands you can train him to trust you and your hands through positive reinforcement. There are great YouTube videos on training parrots through positive reinforcement, and I also have found
Barbara Heidenreich's first DVD on parrot training basics very helpful.
The other thing I'd say to new parrot owners is that parrots are really part of the family. They really enjoy being a part of the activity and routine. They are like children who never grow up and leave home.