Unlike Holy Week and Easter, where the liturgical cycle is greatly expanded and "specialised", the Nativity season follows the standard forms of liturgical practice. Christmas Eve vigil, if it falls on a day other than Saturday, consists of Great Compline and Matins, rather than the more common Great Vespers and Matins (the same happens for Theophany and Annunciation vigils). Christmas Day Liturgy is still essentially the Liturgy of St John Chrysostom, with the antiphons and troparia appointed for the feast.
As for gift-giving, there are no firm traditions I am aware of among either the Greeks or the Slavs, though, in western societies, it is not unusual for gifts to be exchanged among friends. But this is not a church tradition in itself.