Say you don't like your brother in-law. Then you tell your wife you want to go see your brother in-law. Her response: Do you really want to go see my brother? i.e. really = is it true = de verdad Are you sure? Do you really want to see him? Suzy hates studying. "Do you really want to study now?" he asked with surprise.
I love that we are now getting to learn some Spanish-specific lingo. English has Thousands of secondary uses for words into slang phrases. These are some sentences where literal translation might help us to remember, but they will not align perfectly with the way English speakers phrase things. Gotta let go of that expectation.
Yes, it's an adverb, because it's modifying a verb, want. Also, it ends in -ly -- many adverbs are formed by adding an -ly ending onto an adjective. So "the boy is real" -- real is an adjective modifying the noun boy. "The boy really wants the toy" -- real-ly is an adverb modifying"wants".
I answered "Really, do you want to study now?" and it was marked wrong. This is a perfectly acceptable answer that shows that I understood all the Spanish words. Since there is no detailed context, I felt I could place "really" at the beginning of the sentence for emphasis. What a huge waste of time to be taking apart and endlessly discussing these simple answers to reading comprehension questions "does the student of Spanish understand the meaning of this sentence?" The answer is yes...and Duo become more flexible, we do not need to re-learn English here, we are just trying to learn Spanish. Duo you need to adjust your algorithms!!