In physics, esp. quantum field theory, Wick rotation (i.e. putting $t \mapsto i\tau$, imaginary time) is often used to simplify calculations, make things convergent or make connections between different models (e.g. quantum and statistical mechanics). Does the Wick rotation trick occur anywhere in mathematics not related to QFT (analysis, PDE etc.)?
A general form of the Wick's rotation is the "Weyl's unitary trick". This construction allows to relate group actions of noncompact forms of a complex Lie group to those of the compact one by changing the signature of the Cartan-Killing form . Although, the representations of the compact and noncompact forms are different, the unitary trick introduces relations among their invariants and between the transition functions, hence the use in quantum field theory. Also, it introduces relations between their homogeneous spaces (see the example above of the sphere and the hyperboloid).
One instance of (something similar to) that is the construction of models of hyperbolic geometry using a sphere of radius $i$.
Yes. It's called analytic continuation.
Well, since you are looking at it like a "trick", I think that it may be suitable to mention that it can be used to relate the trigonometric functions with the hyperbolic ones (and also to solve somewhat related integrals).