Why did Paul Tankersley accept Summervale's challenge?
Paul accepted the duel because he made a mistake... and because of who he was. The mistake was allowing himself to be provoked into a position which allowed the challenge to be issued "for cause" in the first place. The fact that he accepted the challenge (and met it) was -- IMHO -- inevitable in light of who he was, the information he had at the time, and the consequences (personal, professional, and for Honor) if he had not.
At the time that he and Summervale met, he had no idea (and no way of knowing) that Summervale had a reputation as a professional duelist. For that matter, very few people (in the Star Kingdom of Maticore at large) knew that. It was part of what made Summervale so effective. The reader knew it, because I -- as the writer -- wanted you to know it, so I had Tomas Ramirez and Gunny Babcock explain it to other character's in Paul's absence. Accordingly, all Paul really knew was that a stranger had deliberately -- and successfully -- goaded him into striking the first blow by crudely insulting the woman he loved. (And, I might add, by using an insult which hit Paul especially hard because he knew precisely how hard Honor had found it to open up to him in the first place.)
Paul was the equivalent of a high-level black-belt in a particularly "hard" martial art. Summervale was also a trained martial artist, but Paul had no way of knowing that when he attacked him. Hence, Paul was -- to the best of his knowledge -- guilty of the equivalent (in both practical terms and in the eyes of the law) of assault with a deadly weapon.
The man he had assaulted, however much he might have deserved punishment, had just challenged him to a duel. In demanding personal satisfaction, he was (in the setting of the Star Kingdom of Maticore) renouncing any other form of satisfaction; that is, the duel, if accepted by Paul, precluded Summervale's later filing assault charges over the incident. That, alone, would not have been enough to push someone like Paul into accepting the duel, but it was a factor in his thinking. There was also the fact that whether he had been goaded by the fellow or not, Paul had struck the first blow and, in his thinking (and that of most Manticorans of the time), that meant that Summervale had a right to respond by seeking satisfaction, especially since Paul's blows had drawn blood. Again, social pressure coupled with Paul's own acceptance of that perspective as a Manticoran.
In addition, Paul wanted to take a shot at Summervale on the field. Yes, he knew he had been goaded. Yes, by the time Tomas Ramirez acted as his second, he clearly knew he had been set up by a professional duelist. But at the time he accepted Summervale's challenge, he didn't know Summervale's reputation and did know that he wanted nothing in the universe more than to finish smashing the supercilious, sneering son-of-a-bitch. This, too, was something Summervale had counted on, and it worked. (Don't forget that Summervale was a professional. He'd studied his intended victim carefully before choosing exactly how to goad him onto the field, and it worked.)
By the time Paul knew the truth about Summervale's reputation from Ramirez, he had already agreed to meet him. That moved the entire confrontation to a different plain. Had he declined Summervale's initial challenge, he would have been cut dead by a sizable chunk of Manticoran society, which would have had major repercussions. Although his family was of yeoman stock, it was also extremely wealthy, connected directly to the House of Winton by marriage (remember that he was Michelle Henke's cousin and also a cousin of the Queen herself), and the shame he would have brought upon the family name (and its connections) would, in Manticoran eyes, have been profound. Not only would it have had serious social repercussions for him personally, but it could well have had consequences for other members of his family and even -- to some extent -- on their financial interests. Politically, the Queen's opponents could have used personal attacks on him as an oblique attack upon the Royal Family itself. "After all, if one of the Queen's own cousins lacks the courage to offer satisfaction to a man he viciously beat over drunken words exchanged in a bar, then surely -- given the House of Winton's own notorious temper -- one can hardly put a great deal of faith in the Queen's ability either to think clearly and dispassionately in the present confrontation with the People's Republic or to admit that she might have been wrong and offer the new, enlightened Pierre regime an opportunity to show how different it is from the previous, evil Harris regime." In professional terms, the consequences for his career might also have been profound. For better or worse, military organizations look for officers who are willing -- not necessarily eager, but willing -- to fight and to confront physical confrontations they would not expect/require the typical civilian to face. The fact that Summervale had a reputation in certain select circles as a hired duelist might have been expected to offset that to some degree but Paul didn't know he was one at the time he accepted the duel, and so no subconscious awareness on his part of potential consequences -- social or professional -- was predicated on that basis.
Once the challenge had been issued and accepted, Paul faced a different set of considerations. Yes, by that time he knew that Summervale was believed by some people (like a goodly chunk of the Marine Corps) to be a professional duelist, but he had no proof of that. (If anyone -- including the RMMC -- had possessed such proof, Summervale would have been in prison and not available for any duels.) He had accepted the challenge. To withdraw now, because of his opponent's reputation and record, would have been seen as an act of rank cowardice which would have had even more severe consequences than an initial refusal to meet him would have carried. Of course anyone would have expected him to be concerned, and most people would have agreed that the entire situation was suspicious. But the attitude of the majority of Manticorans would have been that Paul had, in a sense, made his bed. If he only intended to accept challenges from "safe" people and decline to challenge "dangerous" ones, then he shouldn't have been so lacking in circumspection as to punch Summervale out in a bar to begin with. Even leaving aside all of those considerations, Paul himself would have been unable to back down anymore than Honor could have backed down. Summervale had deliberately set out to create a situation in which he could kill Paul. In the process, he had to expose himself to the possibility that Paul might kill him. And Paul was a smart man. He knew, from Summervale's choice of tactics, if nothing else, that he (Paul) probably was not Summervale's only target. He also knew how Honor would react if Summervale confronted her in the same way, and Paul Tankersley was not prepared to protect his own life by hiding behind the woman he loved.
In addition, Paul, as the challenged party, was in control of the protocol chosen, and he chose one in which only a single shot would be exchanged. It was the best compromise between the need to meet Summervale, for whatever reason, and the minimization of the chance of being killed. He clearly understood that he was in greater danger than Summervale, and by the time they faced one another, he knew that his chance of being killed was considerably higher than the chance of his not being killed. But for all of the above reasons, he never even considered not meeting him.
Personally, I thought it was entirely consistent with his character to accept the challenge. I didn't see any need to sketch all of the above out (I catch enough grief over 'infodumps!' ), but it was all present in my thinking and, I'd hoped, sufficiently worked into the subtext of Manticoran society to support the underlying logic of his actions. Please also note that while I do think a case can be made for a code duello serving a useful purpose, I have never been blind to the ways in which such a system can be abused, and the fact that Manticore has one does not mean that I (or, for that matter, Honor) think it is a Good Thing.