Tetraethyl means there are four ethyls attached, you need to list position for every one, hence 4 digits (actually numbers) at the beginning of the name. When you change tetra- to di- there are two ethyls only, so two numbers are enough to describe their positions.
Diagram shows chains of eight atoms, not six.
Double lines (actually triple line) means there is a triple bond there, that's what -yne means.