No, absolutely not. “Grave”, as in “graveyard”, is an Old English (Germanic) root meaning to dig; a ditch or hole in the ground. “Grave”, as in “serious”, comes from a Latin word meaning “heavy”, as in “gravity”. You can find lots of good etymological information here: https://www.etymonline.com/search?q=grave

Yes, it is derived from grave, to mean solemn or solemnly. It has a serious aspect because generally things dealing with death or dying are taken with the utmost seriousness and with sincerity.

Yes, it is derived from grave, to mean solemn or solemnly. It has a serious aspect because generally things dealing with death or dying are taken with the utmost seriousness and with sincerity.