My daughter is highly sensitive - we had to cut out tags and buy specific underwear (or wear skorts without underwear in summer), smell issues (we had to leave a restaurant since she could not bear the smell), it is often too hot or too cold, etc. She is now 13 and doing much better, but at a young age we totally adjusted everything to her, now she is better at adjusting to the world. I would go to a good shoe store and buy her step-in sneakers that are a little big (or replace sneaker laces with elastic yourself, or add to the laces a spring/toggle used on lots of outdoor clothing on the bottom to keep it against your body). The same with a coat, perhaps she could find a poncho-type coat? I saw a kid with a great one, basically a double layer of polartec cut in a large circle with a small circle for the head, the neck was sewn with big loops of a thin strip of polartec and the bottom edge was cut in strips and tied like those blankets you can make yourself - sounded pretty easy to make, It even had pockets inside so the kid could put her hands in). About the gym shoes: could she just bring them and only put them on for gym but leave them in her cubbie or backpack? Our school just requires non-marking shoes that stay on, and my kids have worn those moccasins that have an elastic gore and are pull-on.
There is a great book called "How to raise your child in a world that overwhelms them" by Elaine Aron (Amazon and our library have it) and this is my child - always pausing to check if something is "safe" for her, highly sensitive emotionally and physically but it comes with deep empathy for others, a sense of others feelings, soul-moving appreciation of music and art, and many other wonderful qualities. Sounds like your daughter just had normal little things she dislikes, not the obvious from birth issues my daughter has had. Good luck, love her, give her what she needs now, adjusting to her now will not make her spoiled but will give the confidence that she can function in the world and that will help her make her own way later. Cheers.