Learn English – the meaning of the expression “I can see where you’re coming from”

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.

Your first instinct is correct; this is a noun adjective construction. It does indeed look like the keep something in some condition construction, but that doesn't make sense semantically. It's keep in the sense of retain possession, which is often used to indicate that something else is more desirable or better, as is the case here. The Hogwarts Sorting Hat is the best hat there is, so "you" can keep all the other hats, as they are unnecessary.

Why is the usual word order reversed? Because this is a poem and the author wants to maintain a meter. Black bowlers would make two stressed syllables sequentially spoken, and break the rhythm.