hi fabrice
i think that travelling from mid december to mid january without having any reservations is not a great idea for most places (including south africa) where it is high season and high summer. a lot of people will be travelling, including a lot of locals, and a lot of places will be booked out. also, last minute planning, which doesnt allow a lot of time to plan your trip and to do research on where you want to go etc.
this is especially true as you want to travel around south africa by public transport. if you are travelling by car, its easier to find and drive to alternative accommodations that are available. f you are travelling by public transport, its much more difficult. also getting to or near a national park (i'm assuming you are mainly talking of krüger np) is possible by public transport. but to actually go on a safari and see the wildlife, you need to self-drive or book a tour or day excursions. also it depends on your budget. if you cant find a hostels with availability, but only a more expensive hotel, will that be okay or a problem ?
if you can postphone your trip by a month or two, i would do it. this will also give you time to do some research. if you cant postphone, i would get a guide book asap and look up the places you really dont want to miss. and try to make reservations for them before you leave. cape town if the obvious place to spend new year (and at least 5-7 days in an around the city, as its a fantastic place). but dont go there during new year unless you manage to find a accommodation before you travel.
as for your last question, i'm not sure what you mean. do you mean to spend a couple of quiet holiday days somewhere ? or do you mean to spend a quiet holy day (christmas eve) ?