search in Finder

If you want to delete the saved search, you should remove it from Sidebar before deleting it.

You can do it the other way, but you will likely need to hold command and drag it out of the Sidebar as it won't be able to resolve the alias to the missing search.

Correct. 😉👍🏼

The search folder is saved in a Saved Searches folder in your user Library. Remove from Sidebar merely does what it says, it removes it from the Sidebar.