In order to express a possessive relationship in German between two noun phrases, the usual way is to put the possessor in the genitive case, and after the possessed noun, whose determiner is inflected according to the possessed noun's gender+number+case:
Im Hause meines Vaters sind viele Wohnungen.
Ich suche nach der Spur ihrer Tritte.
But it seems there is alternative, which is for the possessor to come before the possessed noun, taking on the place and role of the determiner for the possessed noun:
In meines Vaters Hause sind viele Wohnungen.
Ich suche nach ihrer Tritte Spur.
Normally, after a preposition like in (denoting position) or nach we'd expect to encounter a dative form:
In diesem Hause sind viele Wohnungen.
Ich suche nach einer Spur.
but I get the impression that the following are incorrect:
*In meinem Vater Hause sind viele Wohnungen.
*Ich suche nach ihren Tritten Spur.
So is it the general rule that if the possessor comes before the possessed noun, acting as its determiner, then it remains in the genitive case, even though more "normal" determiners (i.e. articles or anarthrous adjectives) inflect in agreement with the possessed noun's gender+number+case? Does this only apply in prepositional phrases or also more generally?
[UPDATE: Added examples to highlight the key question]
For example, would the following all be correct (cycling the possessed noun through the other three cases)?
Nominative:
Meines Vaters Haus hat viele Wohnungen.
Accusative:
Ich sehe meines Vaters Haus.
Genitive:
Die Wohnungen meines Vaters Hauses sind viel.