Causality as applied to capacitors
Your first equation is indeed valid for all $t$, but this has nothing to do with causality. You have

$$\begin{align}v(t)&amp;=\frac{1}{C}\int_{-\infty}^ti(\tau)d\tau\\&amp;=\frac{1}{C}\int_{-\infty}^{t_0}i(\tau)d\tau+\frac{1}{C}\int_{t_0}^ti(\tau)d\tau\\&amp;=\frac{1}{C}\int_{-\infty}^{t_0}i(\tau)d\tau-\frac{1}{C}\int_{t}^{t_0}i(\tau)d\tau\end{align}$$

If $t_0&gt;t$ you simply subtract from the first integral the contribution from the interval $[t,t_0]$. But this is still causal, you don't need the future input, you just need to subtract it in case you added it in the first integral. The output could of course be computed without it.

Concerning your second question, you choose the time $t_0$ as the beginning of your processing. The value of $t_0$ doesn't matter but once you choose it, it's fixed. The system is linear if the initial conditions at $t=t_0$ are zero.