Oh, yes. Depends on the avalanche's size (& location & etc.) - but (say) this one is hypothesized to have killed ~1/4 of the at-the-time population of Britain:

https://en.wikipedia.org/wiki/Storegga_Slide

Yes. Close to us, there was a landslide-caused tsunami in Nice, France back in 1979: https://en.wikipedia.org/wiki/1979_Nice_tsunami

I think there was one 100 years ago somewhere on the Canada coast that caused a tsunami and killed some people.

Can also happen in inland lakes: https://en.wikipedia.org/wiki/Tauredunum_event

See: the Forth Estuary.