US Citizen, transiting thru Beijing to go to Mongolia and to North Korea

Hi, We are planning a trip to Mongolia flying Air China thru Beijing from the US. Secondly, we are also planning on taking a tour to North Korea with a travel company based in Beijing. We expect to be several hours in Beijing airport waiting for our flight s to both Mongolia and North Korea. so the question is do we need a transit visa for (1) Trip to Mongolia - same ticket?
(2) Trip to North Korea - different tickets due to tour group arrangements, perhaps even overnight in Beijing. Thanks for your help