There seems to be so much confusion around the UX concept, that the more we talk about it, the less we know. It almost becomes like poetry: a matter of interpretation, as if we don't have a clear idea about it anymore. It's all related to how we feel about it.
It all started with UX simply being the process of designing a user's experience from the very start until the end, with regard to a certain product or service. Theoretically, perhaps this definition is just. In practice, however, there are so many categories of people with so many different interests, that a vast definition like that can only become a battlefield.
For instance, UX professionals will say that hiring managers extend the UX field out of ignorance or in order to pay the same man for numerous tasks. Hiring managers will say that it's the UX professionals' fault, as they cannot reach a consensus with regard to the essential definition of their field. And so on.
Not to mention the unfortunate usage of the term "design" in the context of the rapid UI development, where "design" has a more particular meaning, allowing for a subtle, yet decisive semantic movement. In the end, we all forget where we started from and we end up excruciatingly confused.
That is why when you ask "What should a UX designer know about CSS?', I feel like you ask it out of the lack of a reference point. It almost sounds like a philosophical question. Should s/he?
Perhaps, if we relate exclusively to theoretical backgrounds, we won't be able to answer. We should rely on intuition and on a logical approach.
CSS coding is a matter of implementation, whereas UX is a matter of planning and proactively envisaging the successful interaction between the user and the product. That is why, theoretically, the answer would be "no", just like we shouldn't mix planning with implementation.
In real life, however, understanding the UI stage at the level of CSS coding might be part of the overall picture a UX professional should take into account. Then again, one can get the feeling that a UX professional is bound to take EVERYTHING into account.
In this regard, given that there seem to be no traditional guidelines, we should adapt/select/make the best choices according to our personal profile.
It depends on every person, really. Some of us might be too involved in the abstract notions of planning and might need some engineering activities to bring them back on Earth, why not?
Because of/thanks to its interdisciplinarity, UX allows you to learn innumerable relevant subfields. Let's choose those that complement our journey so far. Let's even change our mindset accordingly.
So, when someone asks us "What should a UX designer know about CSS", instead of wondering what does this question mean exactly, we can regard it as such:
Without the obligation of encompassing CSS coding in our UX education, the manager wants to know if so it happens that our drive so far is consistent with acquiring the said skill.
Then, all the pressure and frustration disappears.