Translation:The engineer goes into the study and works.
In English (at least for me), "workroom" brings up an image of a room filled with tools and supplies for building and making things, maybe with some tables or workbenches. Dolgozószoba brings up a picture of a writing desk, paper, pens and pencils, maybe some bookshelves.
I would really love to understand at last why it is here "bemegy" and it was "mennek be" bust some time ago. What is the rule?
Focus. It's always focus. :)
The focus of a sentence is (usually) on the word that is right in front of the verb stem. If you have a verb with a prefix, like bemegy here, the focus is on the verb itself, that means it's the action that is emphasised. This is what you usually do if you just want to say what's happening.
If you want to focus on something else, you need to put that item in front of the verb stem. That means that the prefix has to be detached and placed after the verb. For instance, if you want to say that it's the engineer who goes into the study, and no anyone else, you can do so by using the folowing sentence: A mérnök megy be a dolgozószobába és dolgozik. Prefix is disattached, so the subject (the engineer) is emphasised.
I don't think so. "Study room" sounds like it's supposed to be a room for studying, but dolgozószoba literally is "working room", i.e. a room where you do (productive) work, not really for learning.
The Oxford Dictionary defines "study" as "3. A room used or designed for reading, writing, or academic work", which covers it pretty well. I'm not sure what your interpretation of "study room" is.