DragonsDoHiccup not much advice but just wanted to say I hope things will get easier for you. My dd is pretty quirky and I always felt there were more issues there. As she got older she mostly calmed down and coped better. She was diagnosed with dyslexia a few years back and knowing that helped.
Please do not give up getting her the help you think she needs but do also enjoy the bits that go well and try and not worry too much.
Some of the things are normal (IMHO) e.g. not wanting to write at 3, and some might simply be you are more patient (looking through the window at the party) - some mums or dads might have given up long before an hour and gone home or might have bundled their little ones into the hall despite protestations. Lots of people think it is easy to get kids to do stuff by badgering, bossing and carrot and stick! But when you meet a child who really does resist stuff you find that no matter how much you badger you will not get your own way! We had this especially around clothes and I think dd has some mild sensory issues around clothing, nothing tight or uncomfortable etc.
You've got a second child not having these issues so I am pretty sure it is not you! Be confident in your own abilities and PLEASE do look after yourself and your own mental health in all this.
I think one thing to remember is, your child is your child, who you love, whether they are on a spectrum disorder or not, have an issue or a disability of any sort etc, they are still themselves. I needed to remind myself of this when dd was very difficult.
I also feel (IMHO) that one day or week or one action or whatever is not the whole picture and not to stress over every bit. But all the 'bits' together do help you to build up a picture.
Maybe keep a record or diary of behaviour, you could include in that things like food eaten as well to see if any foods (sugar or whatever) affect behaviour and also bedtimes etc to see if late nights have a negative effect.
This will help with remembering behaviour when you talk to professionals because it can be daunting and we can always forget what has happened.
If I were you, I would also record issues as and when they happen, what works or does not, just for you. Remind yourself of successes as well as problems.
For all the things you want to encourage her to do can you try sticker charts? With a small 'prize' you can afford? Extra TV, present from the charity shop, trip to the park etc? We did this for potty training and it worked well.
Try not to stress three is not old to be still potty training, IMHO. I know a few kids who were really not ready at three. You can make clothes easier to dress in and to get off for toileting by avoiding things with buttons or zips/fixers, just pull on trousers or skits and simple tops, not too tight.
For dressing with ds (4) we've tried races to get dressed, mummy against ds and just lots and lots of praise when they get it. You can see as you go along what works out. If she gets dressed herself could you give her star of the week or take her photo and put it up on the wall or whatever.
Some kids will find these things work.
Or at weekend you could say, once we you are dressed then we will go to the shops, into the garden, put on the TV, get out the game etc, but Mummy is only going to help a bit.
Really three is still so little, ds is quite bright but still needs help to get some things off or on and he is almost four and a half. All kids are different and they all feel the need to do things in their own time.
DD was very happy to dress herself at three but my friend's boy, although a year and a half older than my dd, was happy to be dressed by his mum when they were both preschoolers.
My dd is 10 now and I love her more than ever but she has challenged me a lot over the years, pushed me away and made life hard, she is just her, and part of my role as her mum is to get her the help she needs. It is not easy.
Good luck.