No, "River Flows in You" is not an original song in the Twilight series. It is a song composed by South Korean pianist Yiruma. It gained popularity independently of the Twilight series.
River flows in you - Yiruma
A Song for Summer has 288 pages.
Song for the Basilisk has 314 pages.
Song of the Saurials has 315 pages.
A Song of Stone has 280 pages.
A Song of Sixpence has 344 pages.
Regina's Song has 448 pages.
Song of Scarabaeus has 368 pages.
Tailchaser's Song has 333 pages.
Song Quest has 240 pages.
On Wings of Song has 315 pages.