1) but how hard is it to book tickets ourselves once we are in China, Bejing?
In china you can only book tickets for travel from China to your next country. And in Mongolia you can only book from Mongolia to Russia. One other issue, the international tickets for travel from China to Mongolia can not be purchased at the train station and MUST be purchased through an agency making them more expensive. There are alternatives but they take an extra day.
2) What would be interesting steps to take from Bejing to Moscow?
I assume that you mean stops. Where to stop depends upon your interests and how much time you have and when you are traveling. Without that information, I am not willing to make too many suggestions other than to say that I would spend 10-15 days in Mongolia and use the entire 30 days that you get with a Russian tourist visa assuming that you can afford this as Russia is NOT a budget destination.
3) Is it possible to get a visum for Mongolia and Rusia in China? And if yes, how?
You can get the Mongolian visa easily. If you haven't already left home, the Russian visa is most easily obtained at home and almost impossible to obtain in Asia. Hong Kong is a possibility but not a guarantee. Same for Kuala Lumpur. Beijing is almost certainly not possible and same for Shanghai. Read the links in my signature line for more information about how to obtain a Russian visa.
4) Am I correct to asume that we can get our China visum (30 days) once we arive in Hong Kong?
Check with Forever Bright Trading, an agency in Hong Kong. Their website isn't great but send them and email and they will tell you what is and what isn't currently possible. There are several EU countries for which it is difficult to obtain a China visa in Hong Kong and in fact, outside the country of your passport. Not sure if Dutch citizens are among those who have difficulty in Hong Kong.
It is especialy hard to find information on how to get a visum for Mongolia once your in China.
Have you tried searching the TT? There are lots of posts about the Mongolian embassy in Beijing, its location, hours, processing times, etc. Cost is is usually 495 yuan if I remember correctly.
Hopefully you have alternative plans as obtaining a Russian tourist visa may be impossible. Transit visa is possible in Mongolia. Tourist visa is difficult and costly but sometimes possible in Mongolia.
westwood - you aren't wrong. Getting the Russian tourist visa in Hong Kong is a possibility. Elsewhere is highly unlikely.
Ruth