Good discussion.
You post a very good question.
"¢ "with" modifiers are very versatile modifiers. They can modify either the preceding clause or the preceding nouns. What they modify actually depends on the context of the sentence and the wording of the modifier itself.
In this sentence, with modifier actually modifies the preceding noun. With modifier in this sentence has the following sense
India's poorest state, which has an annual per capita income of $111
Notice how "which has" can be understood to replace "with".
However, this modifier can be understood to modify the subject of the clause as well because of the nature of the verb - is. This is a linking verb, which establishes the following relationship:
Bihar = India's poorest state.
So whatever modifies India's poorest state also modifies Bihar. You can notice similar behavior of the verb-ed modifier in OG12#5 - Diabetes ranks as...
Now lets consider a scenario in which "with" modifier modifies the preceding clause.
This is a GMATPrep Question. You can find the detailed solution of this question at
this link.
Visitors to the park have often looked up into the leafy canopy and seen monkeys sleeping on the branches,
with arms and legs hanging like socks on a clothesline.
The highlighted modifier above modifies the preceding clause. In essence this sentence can be written as two separate sentences:
1. Visitors to the park have often looked up into the leafy canopy and seen monkeys sleeping on the branches,
2. Their arms and legs are hung like socks on a clothesline.
So sentence 2 has been converted into with modifier. This modifier
extends the thought of the preceding clause by providing a detail supporting it.
Now if the above sounds very complicated, then do not worry about it. As long as you know that these modifiers are versatile and hence can modify preceding clause and preceding nouns, you would be fine. Let the meaning of the sentence guide you. You should understand the meaning of the sentence and ensure that one of these roles fit well.
"¢ Coming to your second doubt about comma + adjective. Good question again. And I believe you have answered this question on your own. Here is how: The correct sentence with choice B is actually a derivative of the sentence with Choice F as suggested by you:
Bihar is India's poorest state, with an annual per capita income of $111, which is lower than that of the most impoverished countries of the world.
Typically this is how such modifiers are derived. As you can see they are derived from which modifiers, and hence they modify the closest noun instead of the far far away noun - subject of the clause in this case. Furthermore, you yourself indicated that "lower" is an adjective. An adjective cannot modify a clause. It can modify nouns or pronouns only.