Little Curly

You are right! Кудрявка is the name of dog. Perhaps, her hair curly)) Кудрявый - curly.

Кудряв[ая] - adjective, 'curly'. -ка - feminine suffix, used either as 1) as English '-er' in, e.g. goalkeeper. 'a person/a thing that does something' 2) to form diminutive nouns 3) to form female gender of nouns applied to males. As "коммунист"/"коммунистка". Probably 2) and 3) are relevant here. [1) would mean 'one who makes things curly' which is hardly the case:)] Usually -ka isn't used with adjectives. So it isn't a proper Russian word, but an 'invention' of those who named the dog:) Sounds funny.