For example, the joke about the Chinese restaurant ("I'd be more concerned about what they're passing off as chicken") plays off of the stereotype that Chinese people eat dogs and cats, and the “passing off” remark implies that the Chinese restaurant owners are deceptive and would immorally and illegally serve their guests a different kind of meat than advertised. I can definitely see how that joke would be considered offensive.
The author labels that joke as "harmless" but you don't have to be a Chinese censor to interpret it as reinforcing harmful stereotypes. I dare you to show that scene at a liberal college and notice how few laughs you get.
Similarly, the racist remarks about Chinese people made by Sheldon's mom are somewhat offensive if taken at face value. I guess the joke is supposed to be at her expense instead ("old people are racists" is an American comedy cliche, if a somewhat tired one) but it's conceivable that either the censors didn't get that, or they feared that their audience didn't get that, so they decided to cut it out entirely.
"They wouldn't get that" is probably also the right explanation for censoring the joke about Jews eating at Chinese restaurants during Christmas, which is a very American tradition. That doesn't imply the joke needs to go, but I can see how that would, at best, leave Chinese viewers scratching their heads.
So, as a German, should I be offended because of the squirrel/rabbit thing? Should Texans be offended? What about the career over partner theme, is that insensible to Germans divorcing due to career-induced burnouts?
No, it's just a joke. I don't believe anyone would think we ate squirrel, and I don't believe Texans do. (However, rabbit is in fact eaten around here. It's also a meat in France (who are famous for their cuisine) and... China. Says the Internet. But around here rabbit is more a delicacy, often for Easter or other special occasions; personally I think I haven't eaten rabbit meat in nearly a decade. Also, the rabbits-for-eating are large animals, not bunnys. Those are adored and loved as pets).
https://www.youtube.com/channel/UCdAi5Y8DDoNyX-4qcEcd-5w
On the other hand, there's apparently a problem where pet stores are selling similar giant guinea pig breeds as pets, but they're too wild and don't have the temper to enjoy it.