Dear NT:

You wrote:
> A-Is this sentence grammatically correct:
>
> 1-The work WOULD take three weeks if we START on Monday.
>
> Some people tell me it is, and some tell me it isn’t. If it is not
> grammatically correct, is it a commonly used ungrammatical structure?

This is fine. “If” clauses are one of the most common reasons for using conditional verbs in English.

>
> B-What is the difference between:
> 2-He would help you if you asked him.
> and:
> 3-He WOULD help you if you WOULD ask him.

#2 is a simple statement of fact–when you ask him, he helps. #3 is a problem of will–he helps when people ask, but you are so far unwilling to ask.

“Would” originally was the past tense of “will” in the sense of “desire.” It still suggests that in uses like this.

>
> Does 3 mean that your asking him is unlikely because you are unwilling to
> ask him?
>
Exactly.