Short answers, pulled from the story.
Frodo Baggins fell asleep after the willow cast a spell that made him feel sleepy. The great tree then tipped Frodo into the stream below while Merry and Pippin were trapped inside cracks within its bark.
Tolkien did not arrive at the malevolent Old Man Willow character until some years later than 1934. Manuscript B dated 1943 shows Tolkien linking a tree with a non-incarnate mind imprisoned within it.
John Tolkien suggests the sketch was based on one of the few unpollarded willows on the River Cherwell at Oxford. The University Parks there hold trees similar to the one depicted in the artwork.
The name Withywindle connects to Old English wiþig meaning willow or flexible twigs twisted into wicker baskets. Windel-treow translates to the basket-maker's tree from ancient language roots.
Trees in Tolkien's world are subject to the corruption of the Fall of Man according to their analysis. Tolkien was a Roman Catholic who believed living things such as trees were affected by original sin.
Old Man Willow was not included in Peter Jackson's films of The Lord of the Rings. The extended edition DVD of The Two Towers includes a scene with Old Man Willow in Fangorn Forest rather than the Old Forest.