To answer your second question, EF classes are marked as "partial" which means you can create a new file (in the same assembly), put another "partial" class with the same name as the generated class in it and the compiler will act as if the code in your new file is present in the generated code.
The first question is more complicated, I would put the business logic where it makes the most sense. If it pertains to a specific entity, make it part of the partial class mentioned above. If it would have to interact with multiple entities, deal with other sources, etc, it should probably go in its own class. Duplicating code is almost never the right answer.
Note that in my opinion, the "Model" is the underlying business logic and data in MVVM. It doesn't necessarily mean that it is just in the data objects, or all in one object. It is simply separated from the View and ViewModel objects.