Is it legal to grow a tree to lvl3 and harvest on the same turn, or would this be two activations of the same space?
It is not legal. There is nothing that violates the principle you mention: a space can only be activated once per turn.
Mentioned in another thread but worth repeating: if you spend an action to plant a seed, both the space where the tree is located and the space where the seed is planted are considered activated with that one action.
In sum, only one thing can happen on a given space on your turn. There are no exceptions.