"There are a lot of people in Tokyo."
It is not the same thing. Yours is the natural way of saying this sentence. The official answer is weird...
Reading of the kanji should be accurate to the context used (人= "ひと" instead of "ジン"; not even including the missing particle "に"). It's not the best for uninformed beginners as it can create problems in the future. Please fix it in order to avoid misunderstandings.
If you see the translation above, there is no に there. Although the translation is a weird sentence by missing the particle に when indicating the location of existence, it is still correct. (I flagged it as being unnatural)
No. 東京の方 does not make sense. Even 東京の人 is unnatural. It should be 東京に人がたくさんいます. は and が follows the usual "shall I mark something as topic" decision.
Yes, it is possible to use あります here, but I don't recommend this especially in beginner's course, because every case can be replaced with いる and it is very easy to misuse ある.
There are cases where ある can be used with animated subjects. One of them is when describing an ownership relation. Like this one, Tokyo has a lot of people. We can use ある to stress the ownership relation that Tokyo owns its people, instead of existence of the people.