How the PCB-A are designed?
Typically everything that can be simulated is simulated. Specially power supplies and everything that is analog. The rest is designed by reading the datasheets and application notes very carefully and following them. 

Using a bread board is quite rare actually. Building test circuits on veroboard is slightly more common. You can use SMD components with those. They can be used to test some specific circuit, which can't be simulated for some reason.

Big part of the design work is just reading the datasheets, and relying on them. With experience you learn to know how things work. Projects typically have an experienced lead designer who knows what to do. Junior designers of the project "obey" the lead designer and learn from him/her.

Peer reviewing is also important. When a design is readyish, it is passed to colleagues who check what you have done and try to find errors or something that can be improved.

Then there are things like EMC design, RF design, etc. The project might include specialists for these areas, that take care of them. Big projects are very much team work.

Here is one more detailed description https://www.radio-electronics.com/info/electronics-design/development-process/tutorial-basics-summary.php

The details of the design process vary a lot. The design houses have different processes, and it also depends on the type of the project.

Then answers to your specific questions:


  Is the functionality split and tested separately?


As much as possible, yes. You can't split everything, e.g. you can't test a digital accelerometer without connecting it to a CPU and actually running code that uses the accelerometer. Most of the analog stuff can be tested in smaller parts, but you still have to also test it all together to know it works


  How about SMD technology ? How do you test SMD circuits without a breadboard ?


As much as possible using simulations. With veroboards it is also easy to solder SMD components.
Mostly everything is tested with a prototype that is built on an actual PCB


  How big designs begin ? Starts with a breadboard ? I hope I am not silly.


The very first thing is specifying requirements for the design. Then simulations, etc.


  If I want to design a power supply (just an example) in SMD technology, where I can start ? To test first with THT components ?


With simulation.