*I don't know spanish, but i looked up the online dictionary.
'to baby someone' = is to treat someone like they are your child.
e.g "Don't baby him" = 'Don't treat him like a child/ baby, treat him like an adult"
'Mimarme' (in the dictionary) means 'to spoil' or 'to pamper'. These are not bad things. 'to baby someone' is seen as a bad thing, and has a negative association.
"I'm peachy" = 'i'm fine'. HOWEVER, it depends on how 'i'm peachy' is said. Usually, I hear 'i'm peachy' when the speaker is being sarcastic, and is in fact, not okay. In this context, the speaker has a nose bleed which is uncomfortable/ painful, so he replies 'i'm peachy' to sarcastically imply he is not okay. Other popular phrases that work in the same way are:
"It's all sunshine and rainbows"
"I'm just dandy"
"jeez, thanks for noticing"
These are all sarcastic replies when you want to imply the opposite. I know this sounds confusing.
I hope this helps a little. If you have a question, feel free to ask me. I'm not quite sure i have explained this well.