correct?

English as a Second Language English Vocabulary, Grammar and Idioms
1
  1. I am in a hurry.
    1a. I am a bit in a hurry.
    1b. I am in a bit of a hurry.
  2. I’ll date her out this Sunday.
    2a. I’ll take her out on a date this Sunday.
    Are the pairs correct?
    Thanks.
2

  1. I am in hurry
    I am in a bit of a hurry.
    1a is incorrect.

  2. I’ll date her this Sunday.
    I’ll take her out this Sunday.
    I’ll take her out on a date this Sunday.
    Sentence 2 is incorrect.

3

I am in a hurry.

Is “in a hurry” an adverbial prepositional phrase or an adjectival prepositional phrase here?

No matter which is the answer, why “a bit” is wrong in the sentence?
It is an adverb and logically it can modify an adjective or another adverb, right?
I am a bit in a hurry.

4

Beeesneees,
Please explain in detail for the question raised by Screen?
Thanks.