Determining time variance of integrator
This is not correct. First, you have to write it like that:

$$
\begin{align}
y(t) = \int_{-\infty}^t x(\tau)d\tau
\end{align}
$$

I.e. your variable in the integrand is different from the integral limits. Accordingly, your calculation will be different, as you will only replace $t$ with $t-t_0$ and not $\tau$.