1) A theory has action at a distance if there is some form of an effect (e.g. a "force" in the classical sense, or an operator in the quantum sense) that depends on two positions in space.
2) It does appear in the dynamics of the theory. If the dynamics of a certain point in space depends on some other point in space, then the theory supports action at a distance. For example, in Newton's Second Law, the force F is caused by some source at point "x" while it causes an acceleration at some receiver at point "y". Thus we have an effect that depends on two positions in space.
3) The force law of the interaction, as described above in 2) above, does support action at a distance. When you write F as proportional to 1/r^2, we are actually defining F as a force created by some source at the origin of our coordinate system. This definition hides the fact that F supports action at a distance (a.k.a. non-locality). We can remove this hiding by instead writing a more generalized F, in which case it is proportional to 1/(r-r')^2 where r is the position of the receiver of the force and r' is the position of the source of the force.
4) Just because something is instantaneous doesn't mean you can't define a "before" and "after". Causality still as meaning here. For example, we can define "before" by saying that the cause of the force F occurs before the source of the force is created. Of course, such a thing is not possible because F is instantaneous, but that doesn't mean we can't define the terms. Really the notion of causality is only useful once you allow for space and time to be mixed together.
There is more physics here then I represented, but hopefully the explanations get to the core of your questions without adding unnecessary complications.