"Yes, my daughter is even smarter!"
Translation:Ano, moje dcera je ještě chytřejší!
"Bratr má delší nos": There the possessive seems to be understood and not needed. Is it also correct to not use it in this situation?
It is about half and half here. A lot depends on context. Unless we spoke about my son in the previous sentence and thus it is understood that the daughter i am talking about is also mine, I would use the possesive here.