Does $ Ev(\cos(4\pi t)u(t)) $ signal has period?
It is a matter of convention. The answer depends on the definition of the step function $u(t)$. In signal processing, it is common to define $u(0)=1/2$. Under this definition, we have that $Ev(\cos(4\pi t)u(t)) = 1/2(0.5+0.5)=1/2$, so the signal is indeed periodic.

If $u(0)$ is defined as either 0 or 1, then you'd be correct and the signal wouldn't be periodic.