1. This is the double possessive form: a possessive pronoun Ďmotherísí & a prepositional phrase with Ďof (my motherís)í. Itís used to mean one or part of a larger group. The cousin you met on the train is one of the few your mother has. If she has only one, you would say she turned out to be my motherís cousin.
2. I think in this context Ďget talking toí could mean Ďstart talking toí. I happened to get talking to means I had the luck or fortune to start talking to. You may substitute "I happened to get talking to" with "I happened to talk to" with a very slight difference in meaning.