What file need run? Answer: SQLyog.exe or SQLyogEnt.exe!
You cannot run 'part of it'. SQLyog is a compiled binary. It does not run scripts in the background. The functionalities of SQLyog are GUI functionalities of the program itself. It does not execute external code. The code that is executes is compiled in the SQLyog binary.
Exception is the SJA ('SQLyog Job Agent'). SJA functionalities can be called from command line or anothe application. Read about it here: http://webyog.com/faq/27_114_en.html
So for executing from a C# program the 'backup powertool' could be a solution if I understand you right.