In poker, at the end of each hand, the winner "lays their cards out on the table" to show they've won. Since part of the game of poker is to try to disguise your hand from the other players, when the winning hand is laid on the table, this reveals facts that previously have been hidden or even deliberately lied about (through bluffing) by the players.
This leads to a common idiom, "lay your cards on the table" to mean showing your actual capabilities. In a negotiation, this idiom can mean revealing your actual wants and what you're willing to give in return.
The idiom is also used in a more general form, "lay [something] on the table" and with a broader meaning, to reveal things that are hidden or simply to speak plainly.
In your examples, there are people who might not normally reveal their true beliefs, and the writers are using the idiom of putting them "out on the table" to mean expressing those beliefs openly.
"On the Table" also means for something to be subject to discussion or consideration by a group. In the written example you provided, the characters views about premarital sex were "on the table". I would infer this is not a subject matter they speak often about.
Rule of thumb: Any English word can be pressed into service in any syntactic role. The only limiting factor is semantics.
Bird, for instance, has an established use as an intransitive verb meaning "engage in bird-watching".
While he was at Harvard, his passion for ornithology flourished; he birded with noted ornithologists Ludlow Griscom, William H. (Bill) Drury, Wendell Taber, Allan Cruickshank, Chandler Robbins, Charles Foster Batchelder and others in the Nuttall Ornithological Club. He met his future wife, Elizabeth Wasson (daughter of Isabel Bassett Wasson), who was also a birder, at the Audubon Nature Camp in Medomak, Maine in 1941. -source
That of course won't support the shit out of it, which has to be transitive; but bird might conceivably be extended to transitive use by speaking of birding a particular territory, meaning to thoroughly observe the bird population.
(It also seems that bird is coming to replace the verb birdie in golf; so you may someday encounter "he birded the shit out of the fifth hole".)
Door has an established use as a transitive verb, meaning "injure a cyclist by opening your car door as he passes".
A Staten Island Powerball winner might have to pay a chunk of his fortune to the cyclist he doored with his truck in Park Slope last month. - source
I see no reason why a deliberate and very harmful act of dooring might not be described as "dooring the shit out of a cyclist".
Best Answer
Many proper and common nouns in English could be used as a verb. One simple example is
The sentence in your example could be rephrased to:
First names are usually used among close friends in casual settings. Scarlet doesn't want Stuart to consider her as a friend (as she considers him as a traitor). That's what "Don't Scarlet me" basically means.
The question and answers posted on ELU, “To science the sh*t out of something” will be helpful.