I would like to know what are the best services to research stocks on the different European stock markets, I would like to cover all EU member states, if possible.
I'm not familiar with your resources or work situation, but the Bloomberg Terminal is best in breed for market research/pricing/data on pretty much any security. Works great for European equities. The downside: it runs about $1,500 a month if you have to pay for one on your own.
Besides Bloomberg and Reuters, I would recommend WorldCap, which is not a terminal but a mobile solution to research stocks in the US, Europe and Asia. While Bloomberg and Reuters run about $20,000 per year, WorldCap has a significantly lower price tag.
Disclosure: I have used Bloomberg and Reuters terminals in the past and I am affiliated with WorldCap.