Maybe the different seas and landscapes will provide enough variety to dislodge this burden from his heart, which he’s so obsessed with that it’s making him act completely unlike himself.

Here Claudius returns to the pose of the concerned parent, though likely he has already arrived at his plan to have the King of England put Hamlet to death.

Haply: perhaps.

something-settled: “somewhat obsessive” (Arden Shakespeare).

fashion of himself: his normal habits.

On “brains still beating,” compare The Tempest 1.2: “For still ‘tis beating in my mind…” See also Emily Dickinson’s “I felt a Funeral, in my Brain” (1862):

A Service, like a Drum –
Kept beating – beating – till I thought
My Mind was going numb –

Archetypes are broad character types that most characters fit in some way or form (e.g., heroes, villains, ingenues, schemers, lovers, wise elders).

i.e., was to be whipped and driven off with stripe marks on his body from the lash.

This use of “stripes” is archaic and can be found in the King James Bible, e.g. Acts 16:23:

And when they had laid many stripes upon them, they cast them into prison, charging the jailor to keep them safely…

The play was not popularly received. This continues the theme previously begun, differentiating between what is popularly acclaimed in the theater and what Hamlet regards as quality. His dissident view parallels his regard for the current regime, and the world in general: an unweeded garden, ruled by a satyr, entertained by screeching children.

In his Shakespearean Tragedy (2nd ed., 1905), A. C. Bradley provides an extended commentary on the Player’s speech:

There are two extreme views about this speech. According to one, Shakespeare quoted it from some play, or composed it for the occasion, simply and solely in order to ridicule, through it, the bombastic style of dramatists contemporary with himself or slightly older; just as he ridicules in 2 Henry IV Tamburlaine’s rant about the kings who draw his chariot, or puts fragments of similar bombast into the mouth of Pistol. According to Coleridge, on the other hand, this idea is ‘below criticism.’….

In essentials I think that what Coleridge says is true. He goes too far, it seems to me, when he describes the language of the speech as merely ‘too poetical’; for with much that is fine there is intermingled a good deal that, in epic as in drama, must be called bombast. But I do not believe Shakespeare meant it for bombast.

More here.

As soon as we met, the King of Norway sent officers to suppress the army his nephew had raised, which looked like a preparation against the Polish army–but on further inspection, turned out to be targeting your highness. Grieved at his nephew’s taking advantage of his old age and weakness, the king sent out for his arrest; Fortinbras complied, got scolded by the king, and in short, promised never to raise arms against your majesty again. Overcome with joy, the old king gave him an annual allowance of three thousand crowns and a commission to employ his soldiers in fighting the Poles.

in fine: in brief.

Norway here is used as shorthand for “the King of Norway.”

assay of arms: assay is a term most commonly used in regard to metals. To assay a metal is to test its purity. Thus, to give the assay of arms to a kingdom is to test its quality – implicitly, it’s validity – by means of war. Compare with the single combat referred to in Act 1, scene i, where old Hamlet and old Fortinbras decided who had the right to rule the disputed lands by a similar assay.

Voltimand reports that indeed young Fortinbras had raised an army with the intent of attacking Denmark. Reprimanded by “old Norway,” young Fortinbras–who hoped to avenge his dead father, and is thus a foil for Hamlet–agrees to forget his plan and is rewarded with money and permission to employ his army against the Poles.

The Norwegian ambassadors from the BBC Hamlet, 1980

In other words, his clothing (“habit”) suggested that he was a civilian plantation owner. We will learn more about his background at the beginning of Part II.

Scholar David M. Owens notes that Alabama is a realistic setting for this story, since it was the site of fierce contention over rail lines late in the Civil War. Bierce, however, has made “an important fictional modification in the geography of the setting. Owl Creek is in Tennessee”–near where Bierce’s own Union Army regiment fought in the Battle of Shiloh, and subsequently guarded and repaired railroads in the area. (“Bierce and Biography: The Location of Owl Creek Bridge,” 1994)

Polonius' words here eerily prefigure (SPOILER ALERT) Ophelia’s suicide later in the play. She kills herself after going mad, presumably out of love and grief for her father, killed by Hamlet’s hand. (Hamlet’s vicious behavior toward her in 3.1–“Get thee to a nunnery”–is a possible factor as well.)

Fordoes: does away with, destroys (archaic).

The Arden Shakespeare notes that the study of music was “another gentlemanly skill.”

A metaphorical meaning may be intender here as well: Let him behave as, and speak of, what he will. Don’t direct, but receive.

Recovering, Polonius explains how his cunning plan will come to fruition: Reynaldo’s conversational partner will confirm that he knows Laertes and, picking up on Reynaldo’s hints about bad behavior, report that he saw Laertes behaving in some similar fashion: gambling, drinking too much (“o'ertook in his rouse,” i.e. in his carousing), arguing during a tennis match (the horror!), or visiting a brothel.

https://www.youtube.com/watch?v=ekQ_Ja02gTY
A falling out at tennis, Wimbledon, 1981