Suggested answer:
(1) Need as a noun:
The need for a male heir to the throne of France was now desperate...

Need as a monotransitive verb:
A cow needs at least a ton of hay every winter.

Need as a modal auxiliary:
But more money need not come from formal development assistance.

(2) The formal differences between modal need and monotransitive need are the same as for modal verbs in general vs. main verbs in general. Modal need takes no 3rd person singual "s" in the present tense and is not supported by "do" to form questions or negatives. Additionally, modal need is not followed by the to-infinitive. The following examples illustrate this:

Modal need, 3rd person sg. present tense:
But he need have had no fears ...

Monotransitive need, 3rd person sg. present tense:
'Even a mule needs rest.'

Modal need in interrogative sentences:
Need it be emphasized that this exposition claims to be no more than...

Monotransitive need in interrogative sentences:
Why does it need to be dark?

Modal need in negative sentences:
And he needn't have let me have it if he hadn't wanted to.

Monotransitive need in negative sentences:
It doesn't need any attention.

Modal need followed by bare infinitive:
First, nothing need be settled as a matter of convention...

Monotransitive need followed by the to-infinitive:
We need to be able to contact the plaintiffs.

(3) Root modality expresses "the subject's readiness, obligation, permission, or ability to do something." The modal need can be used to express the root meaning of "obligation":
I do not think it need be...

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Read about need here
Terminology: modality, root modality