When does this rule warn? Please describe and show example code:
Similar to the quote-props rule, where it's possible to set the rule to consistent and consistent-as-needed, I'd like the object-shorthand rule to include these options as well.

When object-shorthand set to consistent, the below should not be considered a problem:

let obj1 = {
    foo: foo,
    bar: 'This is a string',
};

No problem:

let obj2 = {
    foo,
    bar,
};

The following should be considered a problem:

let obj1 = {
    foo,
    bar: 'This is a string',
};

Problem:

let obj2 = {
    a() {},
    bar: foo,
};

The consistent-as-needed is very similar, but if all keys are redundant, the following pattern is considered a problem:

let obj3 = {
    foo: foo,
    bar: bar,
};

The options apply to both properties and methods.

Is this rule preventing an error or is it stylistic?
Stylistic. Personally, I feel the shorthand syntax is great, but mixing it with the longform syntax makes it harder to scan/read code. In my rules, I want to allow the shorthand syntax, but only if it's consistent in that object literal.

Why is this rule a candidate for inclusion instead of creating a custom rule?
I think this is in line with other rules which also provide consistent and consistent-as-needed options (e.g. quote-props).

Are you willing to create the rule yourself?
Yes.