I honestly wonder if Murtagh is going to get in his own way in regard to Carvahall, Nasuada, or helping Eragon train the new Dragon Riders.
He knows that people hate him and while it frustrates him, he doesn’t blame them. He understands why they hate him: his father’s history and his own actions as a dragon rider under Galbatorix.
He's long since accepted of the reality of people's hostility and opinions of him, but it still stings.
The people of Carvahall don’t know who he is as a person, and they might not fully believe what Eragon has said in his defense.
The biggest accusations are his willingness to join the Empire, trying to capture Eragon and Saphira, and kidnapping Nasuada.
Depending how he defends himself, he might give the wrong impression. The people of Carvahall might want to see him contrite, but Murtagh is too prideful to show that to a bunch of strangers who, in his mind, have no shortage of reasons to put a sword through him.
Though it might get to the point where he decides to share his memories, since his words are not enough. (Might have been Thorn's idea.) I don't know how willing he will be to share the details of the torturous horrors he and Thorn have been through. He doesn't like thinking about them, and he doesn't want to agitate Thorn with traumatic memories. Not to mention the shame he feels for his weakness and inability to protect Thorn and undermine Galbatorix.
In Ilirea, he worries about how his presence will affect Nasuada's standing, and worries for his and Thorn's safety. He wants to avoid Thorn receiving physical and non-physical attacks. He's of the mindset that he can take it and handle it, but Thorn has been through enough of that treatment and Murtagh doesn't want him to experience anymore of it.
He doubts and worries about the plausibility of a successful relationship with Nasuada. He knows she needs an heir but he has high reservations about being a father on the ground of not wanting to give his children a miserable neglected childhood. Even if the court is different from when he grew up, their will still be people who ridicule Nasuada for her lack of nobility and voice derision and hositilty for his past and his father's past, he doesn't want those burdens past down to his own children. He may also fear being a parent similar to his mother. Duty pulling him away, missing out on his children, wondering if they know he loves them but also regrets bringing them into a less than happy life. A life that doesn't have the amount of love, care, and attention they need from him. The amount that he wants to give to them.
With Eragon's offer to teach, he still has his awareness of hostility towards him. He doubts he will respected by any of the denizens at Mount Argnor, or any of the new riders. Though, he does suspect Thorn will have an easier time with any new hatchlings than he will with the non-dragon folk. Thorn has dealt his own amount of damage on the Varden's forces, so both of them are wary of ambush or attack.
Really all this boils down his own doubt, his acknowledgement of his past, anticipation of rejection and lack of respect from others. He wants to protect himself from that, as well as protect Thorn from physical and non-physical attacks.
Though, I think it would be a fun plot thread for Thorn to realize that he's stronger than the two of them think. He can handle harsh words, and with the exception of a Dauthdaert, no weapon can pierce him. Also gets reminded that rare is the person who carries a whip with them. He learns to stand tall against his detractors and any weapons they carry, learns when to put his tail down, and grows in courage when warning folk with a growl and a curled upper lip to watch what they say about his rider (and himself, but he's protective of his prideful stubborn independent rider.)