"This shop is always busy."
While I agree with you, having been to France and experiencing the exact same thing at the start (to a worse extent, I'd even argue), it's much better to learn formal speech first then pick up informal speech so you don't end up making any major faux-pas by using the wrong form with the wrong person! I agree entirely that it should still be accepted in the answers though!
Is 賑やか the kanji for にぎやか? There are some examples https://ejje.weblio.jp/content/%E8%B3%91%E3%82%84%E3%81%8B It seems that it's usually used with street 街 or family 家族. I think it means "many people" literally.
Yes, that's the right kanji. But literally, it means "bustling", "flourishing" or "prosperous", according to my dictionary. It's typically used to describe people (either broadly, as in people in the street, or specifically, as in people in a family) because it has bright connotations with energy and activity.
If you mean, "a shop is busy" as in it is full of people, a Japanese person might say このお店はうるさいです ("this shop is noisy", negative meaning) or このお店は混んでいます ("this shop is full", neutral meaning) or このお店は人気あります ("this shop is popular", positive meaning).
Personally though, having lived in Japan for 2.5 years and able to speak fluent casual and formal Japanese, にぎやか is not an unreasonably rare word. I've heard it used in several different contexts.
It’s an honorific prefix (expressing respect, in this case towards the shop owner). In front of Sino-Japanese nouns it becomes ご- (for example ご両親[ごりょうしん] “parents”). It can be a roundabout way to subtly indicate “your…” when talking to a person of respect, but it can be talking about a third person owner as well. And then there are a couple of words which were used with this prefix so often that it pretty much has become part of the noun itself and you rarely if ever see just the noun without お- (for example tea is pretty much always おちゃ and the bare ちゃ is only used in compounds).