Two year old behavior: is this normal?

Sounds like a highly intelligent two year old that already figured out how to get what he wants when he wants it.  At some point redirection does not correct the negative behavior however completely ignoring him when he does that can work.  I would just walk away when they do it again and let them know you will not speak to them until they stop screaming.  All tantrums should receive the walk away response.  

Sounds like my child. Very strong preferences and very strong aversions to sound. They wanted to listen to music all the time, partly because they loved it and partly to block out other noise.

My advice is to take their discomfort seriously. They are not experiencing sound the way you are. Imagine randomly hearing nails on a chalkboard throughout the day and the anxiety/need for control that might cause if you felt you always needed to be on the lookout.

I recommend: 

1. trying to minimize the amount of painful sounds in your home (program your microwave to not beep; vacuum at night; etc.). 

2. offering ear defenders (the non-electronic ear muff kind). 

3. consulting an OT who is an expert in sensory integration. contact Rita Montez in the East Bay or Liddy Joffe in SF and see if they can work with you or who they would recommend. 

Auditory processing issues are really treatable and for my child, just knowing that I was trying my best to help them “protect their ears” gave them a lot of relief. 

Sounds a lot like my kid at that age. Ultimately he was assessed at Summit Center when he was 7 (at our school's request), and the conclusion was that he's profoundly gifted and also has ADHD. I think it's hard to tell at such an early age exactly what you're dealing with, but to me this sounds a lot like a gifted kid. Unless your pediatrician is concerned, I would just roll with it until your baby is a little older and can be properly assessed. Don't worry about "normal" - it's such a wide range and every kid is so different. 

Please look into giftedness and 2E (twice exceptional) sensitivities. A good place to start is hoagies gifted page. http://www.hoagiesgifted.org

I'm not a developmental pediatrician or a psychologist, just a parent who did very similar things as a kid and has kids who act similarly.

I'm autistic and pretty sure both my kids are as well. You sound like you might be describing someone who's hyperlexic (very interested in words/letters) and potentially has some sensory sensitivities, like me and like one of my kids. A lot of people get anxious when their kid starts showing signs of potentially being autistic, but there's a thriving community of autistic adults who are healthy, happy, and proud to be who they are. If this is something you'd like to learn more about, I'd strongly encourage you to look into the neurodiversity movement and the Autism Self-Advocacy Network.

I'm also happy to chat off-list and provide more resources if you'd like more info.

Good luck and thank you for paying such close attention to your little one's needs. <3

Sounds pretty normal to me. Just because a kid can speak and communicate doesn't mean they can control their emotions. Sounds like a typical 2 year old! And my son is super sensitive to noises, always has been, and doesn't have any special needs or anything otherwise. I sorta chock this up to him being a super sensitive kid by nature. My daughter on the other hand, could have siren go off in her face and be just fine, not in the least flustered. She's much chiller overall but still will scream at us when she doesn't get her way. That's just what kids this age do! 

I see your point. On one hand, your kid sounds like a gifted child based on their language development. On the other hand, sensitivity to noise may be a sign of a condition, although I couldn't even start to guess what that would be. You could ask their doctor for a developmental evaluation and advice. Kids are special, but raising them (especially the "special" kids) isn't easy. Good luck and enjoy the ride!

My son can also talk in sentences like that at 21 months, and he is pretty bossy, which seems typical for this age. I would guess that the sensory sensitivity you describe is unconnected to your toddler being verbally precocious.

I do have a friend with a gifted child who was diagnosed with a mild sensory processing disorder. When she was a baby, she would react very strongly to sounds that seemed quiet to adults (a far away barking dog, a truck, etc). 

All behavior is communication.  Get @ the underlying reason for the behavior being exhibited.  

Disregard the "old-school" approach that you ignore the child for the behavior - that generates more dysregulation & possible trauma as their essential needs are not addressed. 

Get advice from developmental therapists who knows & understands sensory-regulation & processing difficulties & can help you put in place a neurological sound pproach.

HTH