Actually, the first competition was subdual damage. The second event (days later) was lethal. The games are officially sanctioned public events and considered suitable entertainment for all ages (sports, movies, tv being the modern equivalent), and deaths in the arena are not considered a crime. Killing and spectacular combat moves in the games is considered an art form equal to any artist or bard. Being raised in that culture, growing up watching these games, knowing your father excelled in them, and constantly viewing public executions just outside your front door (the inn is located right next to the execution square) would make one rather densensitized to violence, and even to see it as a desirable, effective means of problem solving. Dueling (under certain conditions) is perfectly legal, and in fact, encouraged as the gentleman's way of settling disputes so as not to waste the court's time. Assassins and street scum can be hired to take care of one's enemies for a few coins. Justice, like people, can be bought and sold. Slavery and indentured servitude are legal and endorsed. Life is patently cheap, and ultimately no one is safe from slavery, murder or the chopping block. There is no real police force (just some inspectors solving major crimes of those of interest to the rich), only the army, which is used strictly for quelling riots and defending the city from foreign armies. Most everyone must rely on self-help to get out of trouble, form a Mob to exact street justice, or else hire bodyguards and mercenaries. As long as the target isn't wealthy or connected, there is little to no repercussion for any crime. The courts exist for the rich and powerful, and to make examples out of traitors and other major criminals (such as arsonists) who endanger the entire city. Someone growing up in this kind of society would have a very different outlook on right and wrong then someone living today. I don't think we can apply a modern morality spin on this aspect. Compared to his cultural norm, is he that evil?
Killing in the games where everyone knows what they are in for he has no problem with, including losing his own life. Killing outside the games, even in self-defense, he tries to avoid, and feels less comfortable with, though clearly he will do so to protect his own or his family's interests, or to protect his secrets (within reason).
The fact that he entered the games against his father's wishes (using his father's rival to facilitate this) does show a lack of respect for his father, but on the other hand, it was done because he idolized his father and wanted to be more like him, to experience what his father experienced when he was the same age, to be a great "artist" like his father, perhaps even greater (by proving his refinements to his father's fighting style were superior). All teenagers rebel to some extent, but this is a very grave sin in this culture to go against his father's wishes, and it is certainly grounds for disinheritance, or the father could even sell the son into slavery or legally kill him. The father, the head of the household, has absolute authority over his family, over his children and can do with them as he will, even kill them if they displease him; this is the law (more or less akin to Imperial Rome). So he's taken a huge gamble by doing this. He's been very careful to cover it up so far, and while some may have their suspicions, the only ones who could prove it have been sufficiently bribed or otherwise satisfied to maintain the secret. Quite frankly, he became instantly addicted to the roar of the crowd and the challenge of matching his wits against others in a fight to the death... as well as the fabulous prizes to be won.
Anyway, so he feels like he can't tell his father the truth even though he wants to, at least not yet. Besides, putting his father in the know about everything (including the tax collector) could put his father in danger, not just from the inspector's questions, but from the crime family. The less his father knows, the better, or maybe he's just being foolish?
The only way he could get the suspicious woman to talk to him without beating the crap out of her and waking the whole inn was to make that promise. He figured she'd just stolen something from a guest (which happens a lot) and he could get it back from her, slip it back into the guest's belongings and get rid of her, with no one being the wiser. It was all about protecting his inn's reputation. Nobody expects an assassin; the tax collector (seemingly) wasn't that important, and they had no other guests of any import that night. The crime family is known for keeping its word, and remembering its friends and enemies. It just seemed like good business to not antagonize them and try to resolve the situation quickly and quietly, since they might burn down his inn, murder his family, or ruin his business in some other way.
