Does light (and in general any kind of wave) diffract only when the wave fronts are parallel? Like if you did the double slit experiment when the waves were coming from a point source close to the diffraction grating, would the light still diffract?
Yes it does. A common version on the double slit experiment is, in fact, the one you describe. You can, for instance, install a narrow hole with the primary light source behind it. This hole will act as the source for the experiment, and you can put it as close as you like from the double slit, so the wavefront will be more or less spherical.
If you're talking about more complex diffraction gratings, typically with thousands of slit instead of just two, things are a bit more complicated. Diffraction happens no matter the shape of the wavefront, but interferences behind the grating are a mess with non-parallel light. So, for practical reasons, a source of parallel light is usually used.