I'm not sure about typical English majors vs. CS majors, but in a cognitive sense there is quite a bit of overlap in certain areas. Writing and high-comprehension reading involves the deep understanding and manipulation of language. So does programming.
All "language" is not the same. To be reliably and consistently machine executable, a language needs to facilitate manipulations that are logically rigorous. While in a language for general human communication, emphasis needs to be on manipulations that are more fluid and aesthetic in nature. Facility with one does not necessarily translate well to another. Particularly in the English -> Programming direction, which makes it doubly dangerous to assume it does, when hiring for STEM type jobs. Extreme aspies and savants aside, most good programmers can communicate well in English, in the sense they can formulate their intent precisely. While the converse is much, much rarer.
Yes, but CS majors come out of school with some level of understanding about how computing works... not just how to write code.