Difference between zhongshu and zaishu and zhishu?
植树, 栽树

These terms are written words and sound very formal.

种树

This term can be used in both formal and colloquial ways. 种树节 would sound too colloquial, and is not proper for a special day's name.
For 栽秧 and 植树/种树, instead of 植秧, these are just idiomatic set phrases. There's no need to delve into the difference.