Hello, I am not strictly UP, but I do like the principles of it and try to follow them where possible.
Firstly for the discipline question - which is a big one. You have to get your head around discipline as a whole different concept from what you probably currently think of it as, which is a behaviourist model, ie discipline is rewarding good behaviour, and/or punishing bad, and discipline methods range from the overly positive (praise/reward anything good, ignore the bad) to the strictly punitive (expect perfection and punish/discourage anything less) - most parents, obviously, opt for something in the middle, since both of these are extreme and unlikely to work well in all situations.
However the "UP" philosophy takes the verb "discipline" and goes right back to the root of it - "to teach". That means that, effectively, everything you do is discipline. The way you speak is teaching them how to speak - do you model please and thank you, do you try not to shout, do you listen when they ask you things? You can also teach by explanation, reasoning, answering questions, pointing them towards information sources, helping them make the connection between their actions and the natural consequences of those actions (such as not insisting they wear a jumper when it is cold)
It does not mean that you never do anything which could be taken as a punishment - there will be situations when you need to remove them from a situation for their own safety, for example, or that you never do anything which looks like a bribe - this can be a lesson in negotiation in itself. What it means is that you don't use rewards and/or punishments as the motivational tool - you want them to learn that we don't hit people because it hurts people, not that we don't hit because we will get into trouble. And similarly the idea is that if you make good behaviour about getting a sticker, it becomes about the sticker, and not about the behaviour being a good thing to do.
The difference is that if you are imposing on your child, it's not about your child associating the bad behaviour with the bad experience of the punishment/consequence, it's a way for them to either learn something, or make the bad thing right. So if you send a child to their room for some breathing space, but while up there they play with their toys and have a lovely time, that is fine, because the object of putting them in their room was not to create a bad experience for them, but to show them that sometimes it helps to take yourself away from a situation which is escalating. If they make a mess and you get them to help clean up and they have a great time doing it, that's okay as long as the mess gets fully cleaned up - and if it's something tricky like food mess, you can always redirect, too, e.g. let them tip some dried pasta out onto the floor and then practice sweeping it up back into a tin. Let them play with bath crayons and then wipe the tiles clean with a sponge. Let them "help" with some actual cleaning, rather than making a new mess to clean up. It's about getting them to think about the consequences of their actions as a whole rather than just how things will directly affect them.
So when a situation comes up when your child does something which you don't want them to do, the first question you ask yourself is WHY this is unacceptable - is it dangerous, does it hurt someone, is it actually a problem, or do I just think it's a problem because I've always been taught this behaviour is "naughty"? (The behaviour being annoying to you/other people is a valid problem). Then you have to get that WHY over to the child, if they are mature enough to have the understanding. So, for example, if you have a problem with a child not getting ready quickly enough in the mornings but there is no way they will understand "We'll be late" as a good enough reason, then you plan more time into getting ready and perhaps employ distraction techniques. Once they get older, you can bring reasoning into it. DS is 3 now and knows if he is late for nursery, he does not get to choose which letter he sits on on the carpet, because another child might have chosen it first. Which could be perhaps seen as bribery, but what is he learning? If you are late, you miss things. If you are early, you get more choice and more opportunities.
On your other questions, I don't like reward based systems in nurseries and schools. DS's nursery use stickers, but they don't seem to overly rely on them, which I am happy-ish about. I would love to find a school which successfully used UP type methods but I think it is unlikely, mainly because the behaviourist thing is so ingrained, but also because class sizes are large and it's a necessary short cut for the teachers. They just do not have the time to be explaining and thinking about messages and lessons in a way that 30 different children will understand. But I think that a parent, generally, does have that kind of time to spend with their child, even if you work outside the home (I do.)
I did use bribery for potty training, because DS was over 3 and we were desperate, I knew he could do it, but he just didn't want to. Actually though, it didn't work that well, because he was scared of the potty and even an entire twix (I know 
) wasn't enough to override that. Eventually the thing which got him to do it was putting cheerios in the toilet, and making it into a game, and then pretending we needed a wee and "Oh no, I hope nobody gets there first!" - so the bribery didn't actually work at all, but it was annoying as he then expected treats every time!
New foods - no, but we did use bribery/witholding snacks/puddings for a bit when DS was eating only a bite of dinner and then waking up hungry in the night, and not making the connection himself. And occasionally still do now if I feel he has not made a good enough attempt at dinner and then demanding fruit or whatever. I never bribed him to try new foods, though, he just tried them when he wanted them. (Funnily, the food bribery and the potty training bribery has been mainly since DP moved in as they are not things I would have thought of)
Learning to dress self - no, I don't think this would be important enough as an issue. They will not be asking for help when they are 18 :)
Bathing - again no need - have unceremoniously plonked screaming refusing child in bath before and within seconds he was happily playing - if there is a genuine fear, though, you work through it with them (and this did happen when I left XP as DS always used to bath with me)
Hair washing - not important enough to me. Try various tactics including magic bucket, making into a game, encouraging him to try himself as much as possible. TBH I just keep DS's hair short and it doesn't look too bad. Children's hair doesn't get greasy.
Hand washing - would first try making into a game, but also just wipe them when necessary and ignore the screaming, only takes a second!
Sun cream/hat - again, try the modelling, letting him get involved, making into a game tactics first, and then impose a safety rule that you are not allowed out without this on. Same with car seats. Actually DS thinks that the car won't start unless everyone is plugged in. If we are really in a rush and the games are not working, then it is force, because it's a safety issue, but it's a last resort.