Using objective pronouns as the subject of a verb, when is it okay?

I just have a little question about using objective pronouns (me, him, her) as the subjects of verbs.

1)

They were a peculiar couple, him being a traditionalist and her being more open-minded.

2)

Their mother had high hopes for her children's futures, him to be a doctor and her to be a lawyer.

(Excuse my awkward wording in these examples, you get the general idea.)

Why are these types of usages acceptable? Are they proscribed?

They're acceptable because they are not the subject of a tensed clause.
Instead, they are both subjects of untensed clauses -- a gerund and an infinitive.

The subjects of untensed clauses (when such subjects in fact occur;
untensed clauses often lack overt subjects) may be pronouns, and when they are

• infinitive clauses take the objective (him, her, it, them, me, us) form for their subject,
  and
• gerund clauses may take either the objective or the genitive (his, her, its, their, my, our).
  These are, respectively, called the ACC-ing and POSS-ing gerund complementizers.

That's all, really.

It sounds awkward, but this is the correct form of the sentence:

They were a peculiar couple, he being a traditionalist and she being more open-minded.

This construction is correct because "he" and "she" are the nominative absolutes of the participial clauses.

As for the second example:

Their mother had high hopes for her children's futures: that he would be a doctor and that she would be a lawyer.

Your version is incorrect on many levels.