System.Object being the base class

Very easy. Ape inherits from animal, chimpanzee inherits from ape. Chimpanzee inherits from animal too, but not primarily, only through ape. In .NET, if class does not state its inheritance explicitly, the compiler adds IL code to inherit it from System.Object. If it does, it inherits System.Object through parent types.

Correct, C# only allows single inheritence. The System.Object class is inherited implicitly by your Class A. So Class B is-a A, which is-a System.Object. This is taken care of by the compiler so you don't need to explicitly say that Class A : System.Object (though you can if you want).

Look, you can only have one father. But your father also can have a father. Thus, you inherit some attributes from your grandfather. Dog class inherits from Mammals, which in turn inherits from Animal class, which in turn inherits from LivingThing class.