1. 'Hand skin' is a very weird collocation. The whole sentence is actually worded very strangely. We do actually have an adjective to describe this feeling: 'chapped.'
'I have chapped hands' is how I would say it very simply.
2. I'm going to put my daughter in her car seat.
Don't use 'tie', that sounds like you are putting her in the seat using rope or something. I guess you are talking about fastening her seatbelt when you refer to 'tying.'
3 Technically, it's correct but I find it strange to use 'stair' in the singular. I think most people would say 'stairs' to refer to the stairs as a whole entity rather than an individual stair. Point of note, if you are referring to her doing this outside you would say 'step(s)' and not 'stairs.'
4) Your hat is falling down/slipping down = more natural.
6) In meaning, no. 'Amongst' is used pretty commonly in British English, I don't think it's used that often, if at all, in American English.
7) There is no difference in meaning, although as is often the case, the use of 'up' adds a sense of completeness to the action.