In the sentence:
How deep or deeply should I study something?
Which of the two is more appropriate?
In that context, it’s deeply.
Deep is an adjective, so can only be used to describe the quality of a noun:
How deep is the water?
Deeply is an adverb, so can only be used to modify a verb:
Were you thinking deeply?
Deep is often used instead of deeply, to the extent that it is largely accepted without comment. However, I’ve never heard it the other way around. “What a deeply question” is just plain wrong.