很多的 + noun vs 很多 + noun
Basically, 的 can always be omitted for a simple adjective.
But for emphasis of the adjective, 的 is added (so omit it if not).
For the omission of 的, check this link, kind of CN quora (if it is fine to read).
The third sentence is an exception of your description. 很多 comes after 人, so it is (人很多)的地方 and 的 cannot be omitted here to avoid ambiguity.