Learn Hardcore Chinese: Gōngyuán fùjìn yǒu yínháng, ménkǒu chángcháng yǒu hěn duō rén. - There’s a bank near the park, and there are often many people at the entrance.

It’s the existential “there is/are” pattern: place + 有 + thing. It states existence or availability at a location. In Chinese you cannot say “公园附近是银行” to mean “There is a bank near the park.” Use 有 for existence: 公园附近有银行 = “Near the park, there is a bank.”

Do I need 在 before 公园附近? Is “在公园附近有银行” okay?

You don’t need 在. Both are acceptable:

公园附近有银行 (most common/basic existential pattern)
在公园附近有银行 (also used; slightly stronger focus on the location) Neither changes the basic meaning.

Should I add a number and classifier: “有一家银行”?

Optional:

公园附近有银行 = there is at least one bank (indefinite).
公园附近有一家银行 = there is one bank specifically (uses classifier 家 for companies/stores). Add a number + classifier when you want to be specific about quantity.

What exactly does 附近 do here? Can I say 附近的银行?

附近 is a location noun meaning “the nearby area.”

公园附近有银行 = “In the area near the park, there is a bank.”
附近的银行 = “the nearby bank(s)” (definite, as an attribute), e.g., 附近的银行几点关门？ “What time do the nearby banks close?”

What does 门口 mean here? Whose entrance is it?