Is ideal LPF a LTI system? And is it time invariant?
If your system is defined by its time-invariant impulse response $h(t)$, then it is by definition a linear time-invariant system (LTI). The output of the system is the convolution of the input signal with its impulse response. This holds, since convolution is translation-invariant (https://en.wikipedia.org/wiki/Convolution#Properties).

A time-invariant impulse response can be recognized, if it only depends on one time-variable, i.e. $h(t)$. For time-varying multipath channels, the impulse response can become time-variant, and is in this case expressed as $h(t,\tau)$ and the output of the system is given by

$$y(t)=\int x(t-\tau)h(t,\tau)d\tau.$$

However, these systems are more advanced and not often studied in fundamental courses.