Nothing except some extra emphasis, the same as choosing to include or not include the pronoun in possessive constructions.
It is cute. :) But guys can say it, too, it is perfectly fine. You can also say "kösz". "Kösz", I think, is the correct spelling, but say it like "kössz".
Whatever you do, please don't say "köszike". :)
anyuka is the default form for mother instead of the "actual" anya, Hungarian loves to "cuteify" many things, but köszike is "too cute? lol
Want a technical answer, too?
Remember that the -k in ők is just the plural suffix, and it loses that suffix when combined with other words to strengthen their emphasis. You have that with the possessive forms, as dvanclev already said, and with any of the personal suffixes and postpositions: őmellettük, őalóluk, őbennük, őnekik, and so on. The -k basically moves to the end of the word.
@Sedat that's because Ők loses its "k" when used in the 3rd possessive, e.g. az ő könyvük "their book", and this is the same pattern
Why 'temögüled' and not 'mögüled'? Last lesson was for example 'mögülünk' and not 'mimögülünk'.