Tracing GPS antenna on multi layer PCB
Can I place GPS/GSM modem directly below GPS antenna or should I move
  modem to the opposite side of PCB as much as possible? (I worry that
  there will be GSM antenna traces directly below GPS antenna).


In many cases manufacturers recommend removing the ground plane beneath the GPS antenna, if this is the case placing anything else underneath the antenna could change the capacitance. If the GPS antenna requires a ground plane, then it would most likely be ok to place something on the opposite side of the board as the ground plane would shield any electric fields from the antenna.


  Now I selected 25 x 25 mm GPS patch antenna. Should I select smaller
  antenna (18 x 18 mm) to have more space for better placement of modem
  (not directly below GPS antenna)? What is better: bigger antenna but
  modem directly below GPS antenna or smaller antenna but better
  placement of modem?


There is no way to answer this question by only knowing the size. I'd imagine that the antennas have different gains associated with them. Check the gains and size in the datasheet and select the best one for your application. 


  As I understood, in order to get best performance, entire layer with
  GPS antenna must be plain ground. But what about inner layers (2nd and
  3rd)? Should it be etched below antenna or it is better to make there
  ground pads too?


In some cases, yes, in some the antenna requires nothing beneath it. Manufacturing guidelines for the antenna should suggest the proper PCB layout. The guidelines may or may not be in the datasheet, sometimes they are on a different area of the manufacturers website or need to be requested directly.