I got a cube.When i change the tools different manipulators are show rotate and translate for example.
Now,if you rotate the cube using the rotate tool,and then change to the move tool you see that the axis is changed.
So Rotation affects the translation axis and also the shape of the object.The way i see it you can say that rotation is parent of translation becuase Rotation comes before translation,because it affects it ,but translation does not alter anything in the rotation.The scale you might say is the last child of all.So the hierarchy is Rotation>Translation>Scale ....
No, the translate tool only changes direction because you have it it set to be in object mode (for example double click the tool icon and set it to local or world and it wont do that). Don't confuse the tool with what happens inside maya a tool i a sort of a virtual node that temporarily exists to do a job for you and inject values to nodes as such tools canrepresent the data as they please.
in fact transform comes before rotation the order of execution is (im omitting pivots because theres a counter move after each one)
transform rotate then scale Wich is why they are in that order in outliner and the toolbox also