My DD3 used to be like this. She made me drive her to get her ears double pierced yesterday, at age 17. Six is very young -- give your DD plenty of time.
Supermarket shopping -- screeched until I left for the fresh air and no more muzak. I ended up rearranging my life so as not to have to take her grocery shopping. She hated the car too, screamed from start to end of each trip including the trip home from the hospital when she was a newborn.
Fireworks -- had to leave with DD screaming and puce in the face, calmed down to sobbing after half an hour as we walked home.
Films -- cried for days after 'Curious George', and afterwards if she thought about it.
Refused to read Harry Potter books or watch the films. Actually, she refused to read anything until she was about 9, after learning to read at a normal pace when she was younger. The first book she read after age 4 was The Tale of Despereaux by Kate di Camillo, which she loved even though it was quite suspenseful. She also loved books by Sharon Creech -- 'Love that Dog' and 'Hate that Cat' which were gut wrenching, and others. She loved other books by Kate di Camillo too. Books were nice once she got into reading, because you control the images in your head and you can stop reading any time.
Overall though, she preferred non fiction over fiction once she got to her early teens, though she is once again into fiction this year as their English Lit selection is right up her street ('Heart of Darkness'
).
She can pick up a tune and rhythm instantly, sings really well (lungs got such a great workout as a baby and toddler I am not surprised).
She liked gentle films like the BBC Beatrix Potter series and also Veggie Tales (Christian animated series) and children's programmes on tv. She absolutely adored the Curious George series on American public broadcasting. As she got a little older she liked 'Fly Away Home' on DVD (about a girl who becomes mother to a flock of geese) and '13 Going On 30', 'School of Rock' and many more. I agree with a pp that DVDs are a better idea than the cinema as you can pause them or turn them off and there is no panic.
By 13 she was able to watch Schindler's List in school.
She is a sleepwalker and tends to bottle things up and just gut her way through things.