How to design a 10GHz PCB antenna
First of all is it possible to create 10 GHz wave on a PCB? Any clue
  on how to do something like, regarding the circuitry?


If by "creating" you mean propagating and/or radiating, the answer is yes. However, you need to use specific board substrates like Rogers 4000 series, with low losses and controlled dielectric constant up to that frequency. Your garden-variety FR-4 might seriously struggle with 10 GHz.

If by "creating" you mean generating, the answer is yes as well. It's just a matter of component selection and circuit topologies. You can google and find plenty of design examples of oscillators, mixers, switches, etc.

In any case, you have to very careful with lines width (which will determine the characteristic impedance) and with line lengths (due to phase differences and impedance transformations).


  Do I need two separate antennas, one for transmitting and one for
  receiving, or one antenna is enough? How does one read the signal
  coming from an antenna?


No, you don't need separate antennas, although it may be far more convenient to use separates in some cases. If you don't want separate antennas, there are two main strategies:


Use a triport called circulator, that effectively splits the transmitting and receiving RF paths of the antenna. You can then use both link directions at the same time, but using different frequencies.
In addition to the circulator, use a switch to select whether you connect the transmitting or the receiving RF path to the antenna. This is called a T/R module and is widely used in RADAR, where transmit and receive frequencies are almost the same / very close.



  How do I design the antenna?


This question is way too broad... you should google "microstrip antenna", "stripline antenna", "patch antenna", "printed antenna" and the like.

That's just for the topology of the antenna. If you want to simulate any variation, or exotic things you need specific simulators like HFSS or ADS.  

Don't forget to match the characteristic impedance of the transmission line in your RF board to that of the antenna.


  Does the shape of the printed antenna has any effect on the system's
  performance?


Of course, as you would expect. I can change its impedance, directivity, polarisation, crosspolar rejection, etc.