This question is mixed with a lot of speculation and commentary, so I'm going to try to focus on the questions.
Should the limits of one system of set theory be the limits of a student's mathematical world?
No. The idea that a system of set theory should be a limit to mathematical ideas seems like poor pedagogy (as arbitrary restrictions usually are), and is totally at odds with mathematical practice. I've never heard of anyone teaching such a way, but if someone is, I'm skeptical that the benefits outweigh the disadvantages.
In teaching elementary set theory, while refraining from emphasizing historical decisions that were made in theory construction, is there a danger that students may see the mathematical language of set theory not as a resource providing apparatus to help formulate a given idea, but as a framework that restricts the range of ideas that can be considered coherent and meaningful?
Of course it's always conceivable that students will misunderstand something, but I'm skeptical of the value of obsessing about every conceivable misunderstanding that might happen. My experience has been that constantly trying to head off confusion before it happens just leads to students being more confused by all the digressions. Holding and then discarding misunderstandings is a natural part of the learning process; it's more useful to teach with enough dialog so that if students have this-or another-misunderstanding, there's an opportunity to learn that they think this and address the matter then.
In particular if students are taught precisely one and only one system of set theory -- with enough application of it to mathematics that it becomes like their first language, and they know of other languages of set theory primarily from hearsay -- then is there a danger that the set theory they learn could become a bed of Procrustes for the thoughts that sprout from their minds, like arms or legs growing from the body?
I suppose anything is possible, but this seems fairly unlikely. The jump from "ideas can be formalized in set theory" to "all ideas must be formalized in set theory all the time" is pretty big, and I don't see any reason an ordinary mathematics education should lead students to make such a leap. I also don't see any reason that learning other set theories is relevant; for most topics where there are issues of how to formalize them in set theory, those issues are going to persist in other systems of set theory as well.