I have only been once on Krkonoše in summer. I live in central Germany near a mountain with similar weather conditions and have been on top of it in every weather condition. Snow in may, sun in december, ice rain, day,/night etc. It was sometimes tough but never dangerous. I think it's the same with Śnieżka. It very exposed(wind!!) and can be uncomfortable, but not dangerous.
webcam with temperature for snezka mountain in Krkonoše:
http://kamery.humlnet.cz/en/kamery/snezka
http://www.sniezka.karpacz.pl/index.php?lang=2&menu1=1&menu2=3 with phone number and pictures of sniezka/snezka.
The weather in January can be very unchanging, with low temperatures and sun. But there is no guarantee.
You could proof your shoes or take one pair for changing...back at your apartment/hotel
If you go by train:
Fastest would be to go from Prague to Wroclaw via Dresden, then Krakow annd Warszawa. Dresden is also very worth to visit. As well as the towns Bautzen and Görlitz where the regional express from Dresden to Wroclaw stops. schedules: www.bahn.de This train also stops in Wigliniec where you could change to Jelenia Gora near Karpacz.
Tickets in Germany are more expensive than in Czech Rep. or Poland I guess.
Edited by: european83