Did Tucker Smith actually sing Cool in the movie West Side Story?

Yes, he did. In the original play, the songs was performed by Riff before his demise and it was directed at Action, who was the second-in-command of the Jets. In the film, the song was movied to after Riff's demise. This time, the character of Ice, played by Smith, performed it. Noted that the character of Ice was not in the original Broadway play, he assumed most of Action's role in the movie version. Tucker Smith also performed the Jet's Song. Russ Tamblyn recorded his own vocals but because of contractural reasons, Smith's version was used instead.