I'm wondering if putting a "how" before an independent clause somehow makes it dependent.
Example: John questioned societal norms, and how those norms affected the students.
or
Example: John questioned societal norms and how those norms affected the students.
Both "John questioned societal norms" and "those norms affected the students" are independent clauses, right? Yet when I read "how those norms affected the students", I don't think it's an independent clause (meaning a comma is necessary).
In answer to your main question, the placement of how before an independent clause does not necessarily make it dependent, though there usually is some kind of covert dependency, even when this is not outwardly or formally expressed. I will try to clarify the issues in the following.
Consider this example:
How I answer this question will not make any difference to my salary.
Because it is acting as the grammatical subject of another clause, you can argue that it is not dependent on it. However, it does not make any sense without that other clause.
The exclamations
How you love to dance! or How beautiful you are!
are formally independent, but are best understood as derived from longer sentences in which they are dependent. In the case of these examples these longer sentences could be expressed in multiple ways. Examples include:
Let me tell you... You don't know... I'm amazed by...
Questions such as
How are you? How much does it cost?
have meaning independently, but can be seen as dependent on an implied clause, such as
I want to know....
However, we don't have to follow this kind of (traditional) grammatical analysis and look at how as a single word item. This is why modern lexicographers do not appear to be in agreement as to whether to list how about, as in
How about going out for a drink? How about your mother?
as an "idiomatic phrase" (the Oxford Advanced Learners' Dictionary) under the main entry how, or as a separate lexical entry referred to under how only in the form of a "see also" reference or an online link. The former approach is adopted by the Oxford Advanced Learners' Dictionary, while the Macmillan English Dictionary takes the latter.
http://www.oxfordlearnersdictionaries.com/definition/english/how#how__156 http://www.macmillandictionary.com/dictionary/british/how-about
There is no difference here, by the way, with regard to British or American English.
Finally, to address your (implied) question about the need for a comma before a dependent clause: The presence of "and" in your example obviates the need for a comma.The dependent clause has equal status with "societal norms" as the object of "questioned."
John questioned societal norms and how those norms affected the students
