Is it difficult to get admission to a phd in stem cell biology in Europe?
As aid before, the process is very different from country to country. I personally have experience in Germany, Austria, and Switzerland. 


More and more institutions move to PhD programs. Sometimes it is the only way to do a PhD, and Professors are discouraged to hire students without them going through the initial assessment of the PhD program committee. Usually, these programs are easily to find on the web. See e.g. here http://www.vbcphdprogramme.at, or here http://www.biozentrum.unibas.ch/education/phd/overview/
Look for job advertisements at dedicated portals like ResearchGate (http://www.researchgate.net) or http://www.eth-gethired.ch in Switzerland.
Contacting Professors out of the blue might work in some cases/disciplines. In fields or countries were industry pays much, much more than academia, Professors have problems in recruiting talented PhD students, e.g. computer science or generally in Switzerland. They are glad if they can finally put a qualified person who does the work on the grant they already got month ago.