Yes it means sleep very well. You can say 'The lady sleeps soundly' (she always sleeps well) 'The lady is sleeping soundly' (she's sleeping at this moment) We also say 'sleep tight' Night Night, sleep tight. Mind the bed bugs don't bite.

The word "sound" has several different meanings in English because the noun, verb and adjective forms each have different origins. In this example, soundly means "securely". If you sleep soundly, that doesn't mean you snore! You are "sleeping like a rock", or "dead to the world".

No it doesn't mean snoring.