Is system $h(t) = u(t)-u(t-1)$ LTI or not? $$y(t) = \int_{0}^{\infty} x(\tau)h(t-\tau) \ \mathrm{d}\tau$$ 
where $h(t) = u(t)-u(t-1)$.

I have some problem with understanding whether system LTI or LTV just looking the impulse response of the system. Is there any short-cut to identify the system time-variance looking the impulse function?
If a system is described by an impulse response $h(t)$, then by definition it is an LTI system. Because impulse response of the form $h(t) = \mathcal{T} \{ \delta(t) \} $ only exists for LTI systems.