In this case the preposition is determined by the entity to which which refers, ways. We usually employ different prepositions with different sorts of object. For instance:
Mediaeval authors represented themes of their times through established narrative conventions.
Mediaeval authors represented themes of their times by manipulating audience expectations.
Mediaeval authors represented themes of their times with stock stories and characters.
Mediaeval authors represented themes of their times through often very subtle deviations from conventional patterns.
These are tendencies rather than rules; but some of them are very strong. Method and means, for instance, almost always take by:
By this method Chretien achieves an unprecedented structural clarity.
Chaucer's deepest effects are effected by this means.
Way on the other hand almost always takes in:
Mediaeval authors represented themes of their times in many ways.
Chretien structured romantic narrative in a way for which there is no precedent in earlier analogues.
Since in this case which refers to ways, the appropriate preposition is in:
... will explore ways in which ... because authors represent themes in ways
If it were methods, by would be more appropriate:
... will explore methods by which ... because authors represent themes by methods
In general, you do not need a preposition in such cases. #3 is correct: "I have no idea what word to use."
I'm not sure what the general rule is or if there is one -- this is a pretty broad example. I think, though, that you don't need a preposition if what follows "I have no idea" (or similar phrases, of course) includes a verb, but you do need a preposition if it does not.
I have no idea where to go.
I have no idea how to fly an airplane.
But:
I have no idea ABOUT that.
I have no idea OF my own.
I have no idea BEYOND Sally's last suggestion.
Best Answer
The simple answer here is "He went to America". "To" is used to indicate direction of travel, and so you would usually say "go to (a place)" and "come from (a place)".
However, you should be aware that there are other, idiomatic expressions that use different prepositions with the "to go" verb. These have many different meanings, but if you scroll down the list you will see these definitions:
some examples of these idioms:
and many others.