It may help you to understand why your daughter's behavior is normal with this explanation. And I do believe, based on your description, that she is normal. Remember that everything is new to your daughter. She is only 3 years old. Also remember that she has lots of energy, given to her by nature, so that she can learn all this new stuff. Also, remember that her brain is in the process of developing. We are not born with a mature body or a mature brain.
One way of learning is to repeat the same thing over and over. And if repeating it is fun we continue to repeat it. The way we feel when we say it or do it is one type of fun. Getting a reaction from someone else is also fun.
Temper tantrums happen because the toddler isn't able yet to moderate their feelings. The get frustrated, they tantrum. How serious the tantrums become depends not only on the toddler's maturity but also on the reaction that they get. The parent;s goal is to help the toddler learn how to control their feelings. First, by ignoring the tantrum, we are not feeding into the desire for attention. Then by waiting calmly for the tantrum to pass we are modeling calm. Then by picking them up and reassuring them that the feeling of frustration and anger is OK we are teaching them it's OK to be frustrated and angry but not OK to throw a fit. We then tell/show them other ways to express those feelings. We have to do this over and over until the toddler matures and learns other ways of dealing with their feelings. But by ignoring the tantrum we are teaching them that throwing a fit does not get them attention and especially does not get them what they want.
It is good that she doesn't stay glued to the TV and gets bored and wants to get her energy out. If jumping on the couch is not acceptable show her something else to do. Running in circles is normal and acceptable in most homes. I remember being in grade school and running around my aunt and uncles house. They had a center island with rooms around it. Running was great fun.
Talking non-stop is normal too. She;'s learning how to talk. Yes, it's nonsense much of the time. Language is new to her. My daughter was a non-stop talker and still is at 29, Her daughter is still a non-stop talker at 9. Some of us are outgoing and talk a lot. Others are quiet and barely say a word. Both are normal unless the toddler isn't learning to talk. My grandson was the quiet, non-talking sort and we thought it was because his sister did all the talking. Turned out he has a speech disability. Now that he's learned to talk, even tho he's still difficult to understand, he's also a non stop-talker. :)
I don't know about the covering her ears for Happy Birthday. My grandson has sensory issues and he covers his ears nearly every time something is loud or of a certain pitch. I suggest that there is something about the song that she doesn't like. I wouldn't worry about it.
But if you want to find out more, you could ask her to draw a picture about the song. Because she also cried I'm guessing there is an emotional component to hearing the song. The song seems to make her feel sad.
Relax and enjoy your energetic, eager to learn little girl.