Buying tickets in advance or not?

1/. I plan on travelling to China next year (12 Jan-12 Feb) and Im slightly worried about train tickets from Guangzhou-HK being sold out due to demand because of CNY. How likely is this? Should I just buy tickets in advance online?
2/. What's the likelihood of Beijing-Harbin trains being sold out? (they have frequent trains during the winter periods right?
3./ So lets say I arrive in Beijing today. Would I be able to buy 5 long distance tickets at once for different locations (say, Tai'an to Luoyang; Beijing to Harbin, etc)? Or do I have to go to 'special' ticket booths to do this?

Thank you in advance!!