Difference between "underneath" and "under" when we describe an action

"I slipped it underneath the door" implies it stayed under the door. A door is a thing that is not very wide, and "underneath" would be inappropriate.

"I slipped it under the floor" and "I slipped it underneath the floor" would be the same thing, although I would want to use - incorrectly - "to underneath", indicating that "underneath" is a concrete place, whereas "under" is a position. However, I live among non-native speakers who use that construction, so I probably see a non-obvious logic.

"Under the sea" tells me it's in the water, whereas "underneath the sea" is a place below the water.

As for julio's comment, although it's true you wouldn't really say "it's not over it, but under" (realistically) - you can say "it's not over it, but underneath it."

"It's sitting underneath the table" is not stative - it's still an action. Stative is, like "it understands underneath the table" or "it tastes under the table."

Yes, it is correct. Quoting A Comprehensive Grammar of the English Language :

With words of motion, prepositions may express the idea of PASSAGE, as well as destination.
- He jumped over a ditch [1]
- Someone ran behind the goalposts [2]
- The ball rolled underneath the table [3]
In sentences such as [2] and [3], there is an ambiguity. In [3], we can supply either the meaning of 'passage' or the meaning of 'destination'.
<sub>Note: A triple ambiguity might in fact arise,..., more clearly with:
- A mouse scuttled behind the curtain.</sub>

I would say that underneath suggests one thing is "touching" another. For example, the paper is underneath the book. The opposite being on top of: the book is on top of the paper - touching.

Whereas under suggests one thing is not necessarily touching another. For example, the cat is under the table.

The opposite of under being over. The opposite to above being below.

Rather general and open to exceptions but still a reasonable argument.

"I will slip the envelope under the door" is correct. I would say that "underneath" is used when their is a sheltering effect of something on someone or something. like the door is not so wide so it would be inappropriate to use "underneath" here. if ball is below the table and their is a sheltering effect on it we would say "ball is underneath the table".