Interpreting ceramic antenna datasheets
Do I really need to have all these vias all around my GND plane?


The vias make sure your ground plane is low-inductively coupled to the bottom ground plane. So, yes, you'll need a lot of vias, and placing them on the border of the top layer makes to avoid ungrounded parts of that plane.

If you need every single one: can't say. Get an estimate for the inductivity of your single via, and make sure you have enough in parallel that even at your RF, the effective resistance is small enough.


  I have components soldered on top of the ground plane, will it affect the performance?


Then you've broken the ground plane, and that will indeed affect performance. The point is that you need the ground plane as ground plane around the microstrip feed of your antenna, and where it borders on the "copper-free" PCB area. If you ensure that is "clean", you've done well enough.


  I this is a 4-layer board with two signal layers, a power plane and a GND plane, does this affect the performance as well?


Probably?


  Is the 20mm x 40mm ground plane a recommendation, a minimum, or a maximum? If my PCB is bigger and a different overall form, will it still work?


The "19mm, can even be shortened" in your picture answers the question "is it a minimum?" with "no". 

Generally, it's a ground plane. The bigger, the better, always.