Is it possible for the same person to simultaneously be a Member of Parliament in the European Parliament and in a national parliament? Are any people in both parliaments? Or are there regulations that prevent this?
There are no rules to prevent this in the UK and there are examples where people have been in 3 separate parliaments
Not only is it possible, but it has been done. Moreover, some of the most famous examples were also members of local parliamentary bodies as well.
The best known example in the UK was the late Ian Paisley, the leader of the Democratic Unionist party in Northern Ireland. When the European Parliament was first elected, he stood for one of the Northern Ireland seats and won, ending up as a champion of small parties and independent members against the power of the big parties in the European parliament.
For most of his tenure in Europe (1979-2004), he was also a Westminster MP and for part of the time a member of the Northern Ireland assembly as well.