Sounds like you are fighting unecessary battles
With young kids, sometimes avoidance of an issue is as good a way of getting round things as any. If she hates baths, don't do 'em. give her a quick wash, using wipes if necessary, wash her hair once a month if you can get away with it!
My point is that children are not mini adults - you do not have to hammer away, doing something every day, so that they 'get used to it'....they are all about readiness, and stages, and development.....she is just as likely to suddenly love baths, you are not going to raise a dirty girl just because she doesn't like them right now
also, well within normal to still be in nappies at 2. If she is not poo-ing in her nappy, she's not ready to be out of them.
Screaming to not go home from something she's been doing, again SO normal for this age, I know that doesn't make it easier - but is it possible you could be expecting a little much in terms of ability and behaviour, from a 2 year old? Worth re-visiting your expectations, in terms of normal child development?
I'm not saying she's not hard work - clearly she is, but I am saying it may well be possible for alot of the fights/issues to just disappear if you re approach the way you are looking at what she's doing? I don't know a single 2 year old who behaves reasonably, or logically, for instance!
Also another thing that will reduce your stress is to refuse to let her 'embarrass you'....all 2 year olds behave badly, all adults have seen toddler tantrums...it would be more weird if she didn't embarrass you imo. You don't have to 'own' this as a problem to you and your husband. You can't control what your DD decides to do; but you can deal with it, as you already do. No reason for embarrassment. If people see her playing up, and you dealing well with it, they are only thinking what good parents you are, (or perhaps what a handful she is!)