"My mother's sisters do not eat chicken."
Translation:Die Schwestern meiner Mutter essen kein Hähnchen.
You actually used the genitive structure correctly. A dative structure would be "Die Schwestern von meiner Mutter ...". Note that for feminine nouns like 'die Mutter', the genitive endings are identical to the dative endings anyway . Your problem is that 'essen Hähnchen nicht' is not as good a translation as 'essen kein Hähnchen'.
Yes. Every language has different ways to indicate possession. In English, we have two ways, adding 's (called the genitive marker) of by using the word 'of' (for example, 'the sisters of my mother').
In German, they can also use the 'of' form, and in fact is is more common in colloquial speech: 'Die Schwestern von meiner Mutter'. Note that 'von' takes the dative, so it is 'meiner Mutter', not 'meine Mutter'.
However, they do NOT have a 's construction. They have what is called a genitive case. This is most commonly used by putting two nouns next to each other, one in the genitive case and one in nom/acc/dative case, depending on what is required by the sentence. BOTH nouns must have an article preceding them. The noun in the genitive case is the one that 'possesses' the other noun. So here, you would say 'Die Schwestern meiner Mutter'. 'Meiner Mutter' is genitive (feminine genitive and dative are the same, but masc/neut aren't). Hence, it is 'My Mother's sisters'. Note that in German, the genitive noun is placed AFTER the other noun, which is opposite to English.
Not sure why it is meiner and not meine. The way I understand this is for possesssive pronouns you take the genitive case root so "mein-" in this case and add what would be the appropriate "indefinite adjective ending" based on case,number, gender (nominative, singular, feminine)...even if I worng with the case though the only one that would result in -er is nominative masculine but Mutter is not masculine.
Any noun that ends with the diminutive -chen or -lein (Mädchen, Hundchen, Fräulein, Kindelein) is ALWAYS neuter. Any noun ending in the adverb/adjective-to-noun-maker -keit is always feminine. For example, the adjective 'friendly' can be changed to the noun 'friendliness' as 'freundlich' to 'die Freundlichkeit.'