Yes, FINISHED is a Scrabble word!

Letter values in Scrabble US: F 4, I 1, N 1, I 1, S 1, H 4, E 1, D 2.

Is it a valid word?

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

come or bring to a finish or an end; others finished in over 4 hours"
finally be or do something
have an end, in a temporal, spatial, or quantitative sense; either spatial or metaphorical
provide with a finish
finish eating all the food on one's plate or on the table
cause to finish a relationship with somebody

(of materials or goods) brought to the desired final state
ended or brought to an end
(of skills or the products of skills) brought to or having the greatest excellence; perfected
having a surface coating or finish applied
brought to ruin

Also: finish up, land up, fetch up, end up, wind up, eat up, polish off.