She sounds a bit like my DS who is now 4.6. He refused to potty train, then when he was about 3.2 or 3.3 i thought we'd just have to go for it as he was starting a pre school and they needed to be potty trained (i know they can't legally enforce this but there wasn't really the facilities for nappy changing). Then when i knew he must need to go i wanted him to sit on the potty and he kept saying no he couldn't do it, i ended up getting abit cross with him and making him sit on the potty, he did a wee and was then so happy with himself, that was a Thursday and by the Monday he was dry all day with no accidents!!
He also couldn't dress/ undress himself and wouldn't really try. It was during the last summer when he was 4 i made up start putting on his own shorts, pants etc, again i had to get abit cross with him so he'd try and then when he realised he could do it he was fine. Since starting school he's come on really well with the dressing/ undressing and now will often say he doesn't need help.
He also doesn't like walking but i think a lot of them are like that once the novelty they have when they first learn to walk wears off. I think you just not bring the pushchair then they don't have a choice. Theres no way i would be carrying a 3 year old in a sling!
He doesn't like noise either but is ok with personal space. Again he struggles with changes in routine like your DD.
He has always been really good with his numbers and letters and has an amazing memory for certain general knowledge things like your DD.
When he was younger at his first pre school they though he might be ASD but the health visitor said she thought he was bored at the pre school and i moved him to another one and he got on great. My gut feeling at the time was that he wasn't.
He started school in sept and had a shaky start but is now getting on really well although has not liked the past couple of weeks due to changes in the routine for christmas plays etc
With DS i think alot of things, especially the academic stuff comes easy to him without much effort so whenever he encounters something where he has to make abit of effort (potty training/ dressing) he won't even try. Its almost like when he things he might not be able to do something he's scared he'll fail so won't try.