how do you progress the potential protagonist trait?
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Protagonist, Cat Person, and Party Planner are all based on the 'highest level achieved at death' for Liam, Lani, and the entire party. Those are currently not possible to max out, since enemies don't scale up high enough to earn the required levels (with dungeons, I think the max level of these titles is 7 or 8, and in the free public version, it'll be around level 4).
The chicken dinner flag is worded poorly, so lots of people have that same question. There is an in-story method to kill the chicken before you fight it, and thus you eat it with your family and earn the flag. If you catch all of the chickens in the yard (with Marla), you'll be presented with the option to catch the rooster. If you have the maximum level of the chicken title (Alektorophobia), then you'll kill it. If the title is too weak, you'll get scared and run away.