It is the genitive case (possessive). In the sentence "That is the father of her husband", "her husband" is not the subject nor the object nor indirect object of the sentence - it merely serves to further define father. Not the father of my child, nor the father of her friend, but the father of her husband. When it's the genitive case, the endings can change for nouns, adjectives and articles. Here is a link that describes it.
About halfway down that article you will see where it says "The same endings are used for the negative indefinite article (kein-), and the possessive pronouns, mein- (my), dein- (your, used to a friend), sein- (his), ihr- (her and their), unser- (our), euer/eur- (your, if addressing a group), Ihr- (your if addressing an authority figure, always capitalized). The chart nearby shows the -es ending.
I don't think so although I found this: http://fr.forvo.com/search-de/Mannes/ with the last syllab stressed. However, in other words the first syllab is stressed: "Junges" - http://fr.forvo.com/search-de/junges/; "Sohnes" - http://fr.forvo.com/search-de/Sohnes/. Native German speakers, please clarify!
father-in-law = Schwiegervater. Ein Schwiegervater ist der Vater eines Gatten (maybe)
I didn't find a declension at the link provided by BenUserName below; however, I did find one in the Duden. I think "eines Gatten" is correct.
Thank you for the correction, BenUserName.
If the sentence were "That is her husband." then husband would be the direct object. If it were "That is the father." then father would be the direct object. Am I right? But since the sentence is "That is the father of her husband." that puts husband in genitive and I guess father in nominative?