What would be the difference between "You study in Rome." and "You study Rome."
This could mean the latter, though is much more plausibly interpeted as the former.
You could instead say: "rebus romanis studetis" ("you study Roman stuff") or "urbi Romae studetis" ("you study the city of Rome").
In "Rōmae studētis", the word Rōma (Rome) is in the locative case, which means that something is being done in or at a place or location. For "You study Rome", you would need the accusative case, since Rōma is the direct object (the thing being studied): "Rōmam studētis".
Also keep in mind that "studētis" is used when you're talking to more than one person. If you were only talking to one person, you'd say "Rōmam studēs."
As a general rule, "studere" does not take a direct object, but a dative.
There are some examples of an accusative used with this verb, but they are either from pre-classical writers, or they are neuter pronouns, like "hoc unum/omnia haec/quod studet".
STUDETIS is a mistake. It should be STUDITIS because that verb belongs to the 4th conjugation.
No, it is correct as "studetis". It belongs to the second conjugation.
Perhaps you are misremembering due to the "i" in the noun "studium". I remember making this mistake myself once when writing a letter to Reginald Foster, who definitely let me know about it!
Well, I just have to admit that you are right. Sorry, I have not studied Latin seriously for many years, and I was pretty sure that it was STUDIO rather than STUDEO. I should have checked before posting my remark.