I don't want to speak to much about my specific situation but speaking generally...
When you move to an EU country you have an automatic right of residence as a job seeker. You have to register your presence in the country within 3 month (although taking longer doesn't result in any consequence). It is unlawful for the host nation to routinely check criminal records. Remember, you're not applying for a visa but rather just telling them you've exercise your free movement rights and are now residing in the country. Therefore, they're not going to know about your past unless you've been in trouble in that country.
As for money... I didn't have that problem because I moved in with a friend until I found work. Saying that though, I found work teaching English within 2 weeks. Knowing English in is a huge commodity. I earn quite a bit of money teaching English to groups of business people, often during their lunch break. I earn around 15 euros per hour, which for me is fine. The cost of living in many European countries is much lower. Realistically it could be done with just enough money to rent a room for the first few months. Depending on the country, when you register as a resident in the country you have to be able to prove that you have sufficient resources as to not become a burden on the state (their is an exemption for students). Therefore I'd recommend not registering until you have found a job.
Things are completely different in terms of checks in Europe. Most countries do some sort of checks for working with children, for example, but apart from that checks are fairly uncommon. This whole thing about an employer having a right to know if you've been convicted would seem very alien to most legal systems. Criminal records are private. In some EU countries not even the police have direct access to them.