Sorry, but just about everything you said is wrong.
Firstly, her reaction to the joke says nothing for her own work ethic. I shouldn't have to explain this to an oh-so-mature 23 year old (lol) so I won't.
Secondly, most of what you said is anecdotal and self-serving. Just because you are one of those "cool girls" who would laugh at being the butt of the joke is, frankly, meaningless. Glad you don't care, but I'm not sure why OP should care about your assumed cool girl reaction; care to explain why you are the example for all correct behavior?
Thirdly, lol. Culture of the company. Lol. You do know that just because a company's "culture" is X doesn't mean it's right or even acceptable, yes? Wow, this is the worst piece of "advice" in your post. Let me explain why. When I worked at a smaller company a few years back, the humor was quite vulgar. I didn't mind, I'm pretty laid-back myself. But I very distinctly recall a start-of-the-year company event in which we had a nice dinner, played some games, chatted, and discussed the future of the company. It was blatantly said that while the current staff had no problem with the company's present level of humor and communication, if a customer overheard said humor during a call or a future new employee found it offensive, it was something that could literally shut down the company if legal action was taken (lawsuits against smaller companies hurt like a bitch). I am not saying the OP could or should take legal action; unless she is a minority, the jokes were sexual in nature, etc., she probably has no recourse and it's probably not worth it. It sounds to me like this company is humorously small and clueless anyways. I'm just pointing out the fact that just because a company's current staff is fine with particular behavior does not mean the problem is the newcomer, as you mistakenly believe. If they ever take it too far, you can shove your "mature humor" (which it's not, and you're not mature for enjoying) up your ass, along with your company in legal proceedings.
While I'm on the topic of being "mature", as you feel so proud to label yourself: cracking hurtful jokes at the expense of others does not make you mature, smart, or funny unless you know those involved share the same sense of sniping humor. Not many do, not when they're on the other side of the barrel. And in fact, my own dad has this "mature" sense of humor that you enjoy. And I've watched him make many a person very uncomfortable with it. Being hurtful does not make you funny. Period.
You may think this is stuffy but that would, in fact, show you to be the less mature and experienced one. Businesses exist to make $$$ which is why they follow certain procedures, have rules, follow the law, etc. A company this size taking low blows at the new hires is taking a risk.