Grammar is not the science of building fragments of sentences. Grammar is the science of joining words together to create full sentences.
The fragments you provide in your questions are all incomplete. They can exist in German sentences, but they are not closed units.
Here is a full sentence:
Der Hund meiner Schwester ist schwarz.
The dog of my sister is black.
The word »meiner« follows »Hund« immediately, but still the two words don't belong to each other. The part »der Hund meiner Schwester« is a nominal group which has the role of the subject in this sentence. Because it's the subject, the whole group is in nominative case, which means that the core of this group and its determiner must also be in nominative case. The core is noun »Hund« and its determiner is the definite article »der«. So, insider this group the words »der« and »Hund« belong together.
The part »meiner Schwester« is another nominal group. Its core is the noun »Schwester« and it also has a determiner, but this determiner is not an article, but the possessive pronoun »meiner«. But this whole nominal group (»meiner Schwester«) is a right genitive attribute of the core of the outer nominal group:
[Der Hund (meiner Schwester)] ist schwarz.
Each pair of brackets encloses a nominal group. The 4 words long group in square brackets is the subject of the sentence. It is in nominative case. The 2 words long group in round brackets is the right genitive attribute of the noun »Hund« which is the core of the group in square brackets.
The gender of the possessive pronoun must match with the core noun to which it belongs, so it must match with »Schwester« which is a feminine noun, so the possessive pronoun must be feminine too:
Der Hund meiner Schwester ist schwarz.
Der Hund meines Bruders ist schwarz.
Der Hund meines Kindes ist schwarz.
The possessive pronoun is not connected to the core of the outer nominal group. The noun »Hund« is masculine, but still the pronoun of the inner group must match with the core of the inner group.
Der Hund meiner Schwester ist schwarz.
Die Katze meiner Schwester ist schwarz.
Das Kaninchen meiner Schwester ist schwarz.
What we've discussed here was a right genitive attribute. There is also a left genitive attribute:
Meiner Schwester Hund ist schwarz.
My sister's dog is black.
The outer nominal group (»meiner Schwester Hund«) no longer contains a determiner, because the left genitive attribute now takes this role. The inner nominal group (»meiner Schwester«) is still the same as before. It is a genitive attribute, but now it stands left of the noun who's attribute it is.
Using a genitive attribute on the left side is rare and not very common when this attribute is a nominal group with more than one word. »Meiner Schwester« has two words, so you better use it as right genitive attribute. But when the attribute is just one word, it is quite common to use it as left attribute. This is often the case when this genitive attribute is a name, because names usually are used without articles:
- right genitive attribute:
common: Der Hund meiner Schwester ist schwarz.
not common: Der Hund Barbaras ist schwarz.
not common: Meiner Schwester Hund ist schwarz.
common: Barbaras Hund ist schwarz.
But genitive case is not only used as the case of an inner nominal group that is an attributes of the cores of an outer nominal group. There are also genitive objects:
Irene bedurfte meiner Hilfe.
The German verb bedürfen needs a mandatory object that tells what the subject needs, and bedürfen needs this object to be in genitive case. The nominal group »meiner Hilfe« is in genitive case, but it is not an attribute of something. It is an object, and it is in genitive case, because the verb »bedürfen« wants it to be in this case. And, as before, the possessive pronoun »meiner« is part of the nominal group »meiner Hilfe«, so it belongs to »Hilfe«. There is nothing else in this sentence to which it could belong.
Also in the Russian phrase »собака меня« the two words do not belong together (they are not part of the same grammatical unit), and they do not translate to "der Hund meiner":
- Моя собака меня укусила.
Mein Hund hat mich gebissen.
My dog bit me.
- Почему собака меня не слушается?
Warum gehorcht mir der Hund nicht?
Why doesn't my dog listen to me?
- моя собака меня очень любит.
Mein Hund liebt mich sehr.
My dog loves me very much.
Der Hund meiner ist groß.
into Russian would beСобака меня - большая.
, asmeiner
is the genitive declension ofich
, andменя
is the genitive declension ofя
(it is also the accusative declension, but that is not what I was considering).