No the movie Holiday Inn was made before White Christmas. The movie Holiday Inn debuted the song White Christmas and the Movie White Christmas was made to promote and build on the popularity of the song.
The theme song from the 1963 movie had no singer. There are lyrics by Matt Monro, but no one sang them for the movie. Not strictly true. The song was sung at the END of the film, but it was still the theme tune.
The soundtrack of Gladiator is composed by Hans Zimmer, Klaus Badelt and Lisa Gerrard. The last track of the Album is Now we are Free which is regarded as the theme song of Gladiator. The composer of the song is Lisa Gerrard