How are complicated PCBs designed?
how do engineers calculate the right parameters of components (caps,
  resistors, inductors, ...) in big projects, such as mobile
  motherboards and etc.


They don't calculate. They use highly polished reference designs from individual IC manufacturers and only accommodate/integrate them into their particular space constraints. Sometimes with mistakes and violations, that are corrected later.


  How do they do that? Is the board multilayered to make it possible?


Yes, this kind of boards have 8-16 and more signal layers.


  If so, that's odd because no vias can be spotted in here.


They use "blind" and "buried vias" that are very small and filled during metal plating, and are invisible when covered with solder mask material.


  Or what other kind of techniques are used to design such complicated
  boards?


No special techniques, just high-end CAD software packages. 


  Do they align the components by themselves or is it done by software?


Part placement is dictated by PCB manufacturer's rules and preferable alignment is defined by "pick-and-place" machines used for automated board assembly.


  It feels rather complicated and very time consuming, let alone error
  sensitivity - one mistake and oops.


Yes it is complicated, but CAD tools have all sorts of DRC (Design Rule Checks), and mistakes still do happen. That's why mass-produced devices have certain engineering discipline to bring the board in several steps, refining the design.


  If it's all about several ICs – all the passive components around it –
  do they just follow reference sheets of these ICs?


Yes, OEM usually integrate several large design blocks following reference designs meticulously. And these parts are usually debugged and supported by design support teams from original IC producers.