Yes, EVE is a Scrabble word!

Letter values in Scrabble US: E 1, V 4, E 1.

Is it a valid word?

EVE is a valid word in the official Scrabble dictionary (US), the UK Scrabble Dictionary and Words with Friends (ENABLE).

(Old Testament) Adam's wife in Judeo-Christian mythology: the first woman and mother of the human race; God created Eve from Adam's rib and placed Adam and Eve in the Garden of Eden
the day before
the period immediately before something
the latter part of the day (the period of decreasing daylight from late afternoon until nightfall)