Both forms of each sentence are both grammatical and in common use. Verbs ending in -ing are generally present participle tense, and you can use either the possessive form or the subject form of the pronoun.
If you changed the verb to the infinitive (to verb) tense, you would need to treat the pronoun as the corresponding subject to that verb.
This is easy in the first case:
The position entailed him to be in Chicago most of the time.
It's more difficult in the second sentence:
The bonuses required him to reach the quota.
It's impractical to so with the third and fourth sentences.
In addition, I would recommend avoiding the phrase "assume/assuming the/that position" where possible, as many associate that wording with corporal punishment.
ACC-ing him and thePOSS-ing his are correct for gerund complement clauses. It's a "speaker's choice" situation; there is no meaning difference. – John Lawler Sep 20 '14 at 23:51