// in the first session
Cat cat = (Cat) firstSession.load(Cat.class, catId);
Cat mate = new Cat();
firstSession.save(mate);

cat.setMate(mate);

Use secondSession.update(cat) and secondSession.update(mate)

You should save firstSession reference and call continue() method to restore it

Its impossible, all changes should be initiated and saved within the same transaction

cat.update() and mate.update()

Use secondSession.merge(cat); and secondSession.merge(mate);

You should recreate these objects in new session