Implementing a TI antenna design on an existing PCB
I imagine that you can do whatever you want in the pink zone so long as you keep all the conductors as-is in the area used by the antenna (green/white zone).
As for ground plane the image is a two layer board and the plane under the microstrip (the vertical; part above the socket) should be retained at the same distance to maintain the impedance match.
If you are using more layers the microstrip could be redesigned for a closer ground plane allowing the other side of the PCB to be populated behind the microstrip.
The zig-zag antenna uses both sides of the board so you can't place components behind it and should not place any conductors close to it.