How we cite our quotes: (Book.Chapter.Paragraph).
Quote #4
"It is the will of Théoden," said Háma.
"It is not clear to me that the will of Théoden son of Thengel, even though he be Lord of the Mark, should prevail over the will of Aragorn son of Arathorn, Elendil's heir of Gondor."
"This is the house of Théoden, not of Aragorn, even were he King of Gondor in the seat of Denethor," said Háma, stepping swiftly before the doors and barring the way. His sword was now in his hand and the point towards the strangers. (3.6.36-8)
Both of the named representatives of Théoden whom we've met by this point in Book 3, Chapter 6, Háma and Éomer, are exceptionally loyal to Théoden. The fact that Théoden inspires so much loyalty in these good people makes us wonder about those passing references to Wormtongue and to unjust or needless orders in Théoden's house. Something must be wrong if a man who can inspire such loyalty can also be so harsh and cruel with his orders, as with his demand that the guards only use the language of Rohan to greet strangers, against the "custom in the West" (3.6.18).
Quote #5
I do not lie. See, Théoden, here is the snake! With safety you cannot take it with you, nor can you leave it behind. To slay it would be just. But it was not always as it now is. Once it was a man, and did you service in its fashion. Give him a horse and let him go at once, wherever he chooses. By his choice you shall judge him. (3.6.126)
Gandalf is remarkable for his mercy. He uses almost exactly the same argument on Frodo to explain why Bilbo didn't kill Gollum in his adventures in The Hobbit: "Many that live deserve death. And some that die deserve life. Can you give it to them? Then do not be too quick to deal out death and judgement" (The Fellowship of the Ring 1.2.149). So that begs the question: what or whom is Gandalf loyal to? If he were loyal to Théoden, he might be inclined to kick Wormtongue to the curb. But he seems loyal to some deeper, moral idea, rather than to any one person.
Quote #6
"You come at last, but too late, and with too little strength. Things have gone evilly since Théodred fell. [...]
Théoden had sat silent, hidden from the man's sight behind his guards; now he urged his horse forward. "Come, stand before me, Ceorl!" he said. "I am here. The last host of the Eorlingas has ridden forth. It will not return without battle."
The man's face lightened with joy and wonder. He drew himself up. Then he knelt, offering his notched sword to the king. "Command me, lord!" he cried. (3.7.9-12)
Between a king and his people, there is (or should be, at least) a kind of contract: if the people give their loyalty to the king, he will do everything he can to keep them safe. But as Ceorl inadvertently points out here, the people of Rohan have been suffering because of Théoden's Wormtongue-inspired neglect. He leaves the Riders of Rohan to be slaughtered by Saruman's forces because he is busy feeling old and decrepit. It's also worth noting that the loyal relationship between King and subject was broken not only by Théoden's neglect, but also by Wormtongue's bad advice. As a subject, he's duty-bound to tell the King the truth, but that worm-tongue of his keeps getting in the way.