I actually took this class at Cambridge very recently, so I feel very qualified to answer.

As for your first question, the entire course is taught in the context of the Euclidean signature. This is primarily because it is taught within the mathematics department, and a vast majority of work done in mathematical QFT is done in a Euclidean signature. This way, we don't have to worry about pseudo-Remannian manifolds and the partition function is a bit more well-defined as the action $S$ is bounded from below.

As for your second question: you are correct. Typically, a generating functional contains a linear "source" term. And you are free to add one if you want. However, in the context of the course, this chapter simply serves as an introduction to some advanced topics (symmetries and Ward identities, the graph theoretic interpretation of Feynman diagrams, effective field theory, and supersymmetry) in a context where the topics themselves aren't obscured in messy integrals.

I hope this helped!