Subdomains, problems following Tutorial

Yes because V is the function space of your problem.

The idea behind the DG0 space comes from the fact that the degrees of freedom are defined by a single constant per cell. This coincides with the tags assigned for marking domains using a mesh generator, like GMSH. See, for instance https://fenicsproject.discourse.group/t/set-expression-values-according-to-gmsh-physical-regions/13958/3