Sven, it's perfect. Welcome aboard.
It can literally be whatever you want. Last names aren't uncommon: Remember, there were still people alive when shit hit the fan, and they probably insisted that their kids have a last name. With others, they only have one name, but will sometimes adopt the name of their town as a family name (like for example, someone might have the last name "quartz" if they came from New Quartz). Hell, someone might give their kid a different last name from them (like Thunder if they were born during a storm).
As for clothing, it really depends. One of my characters, because she lives in the desert, wears really ragged and kinda skimpy clothes due to the heat, where as another one wears long sleeves in the desert to prevent sunburn. Some wear cloaks to keep the sun off, where as some in rainy areas will wear a raincoat or waterproof trenchcoat all the time. People are more concerned with survival than fashion or even comfort.
Alright, registering in my two characters.