Frankfurt to Paris?

I'm working on a schedule that involves several family visits in one trip to Europe in September and it goes something like this:

Vancouver to Frankfurt

Frankfurt to Paris

Paris to Berlin

Berlin to Frankfurt

Frankfurt to Vancouver

So I have two related questions:

1. What is the cheapest way to get from Frankfurt to Paris, rail or city hopper flight?

2. Similarly, what low-cost air carriers offer service from Paris to Berlin and Berlin to
Frankfurt.