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I think 植(as a verb used separately, when compared with used in compound verbs like 移植 种植) is somehow formal. Hence except for the idiomatic use of 植树, 植花 is rarely heard but 栽花 种花 is very common.
植物节 is a proper noun and using 种树节 or 栽树节 will make it sound informal.
植秧 seems to exist. However, the following example is from a written record. I have no idea whether 植秧 is commonly used in spoken Chinese.
Nonetheless common use seems to be 插秧 or 栽秧. In fact, 插(insert) and 栽(plant) here refer to 'transplanting'. The way to grow rice is : first grow the seedlings in a nursery and then transplant them into fields. (http://www.knowledgebank.irri.org/training/fact-sheets/crop-establishment/manual-transplanting) 插 describes the action of inserting the seedling into the field. 移(to move)秧 is also acceptable. 种秧 seems to be a dialectal use.
Ancient Chinese used 莳秧 (according to 说文, 莳 means exactly 'to transplant'), which is still used in the Wu dialect.
These terms are written words and sound very formal.
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.