'Could' is technically just the past tense of 'can'.
I can run faster than most people.
= I can faster than most people now.
I could run faster than most people.
= I used to be able to run faster than most people in the past.
But 'could' is also one of the ways of creating the subjunctive mood ... used for things which are uncertain or indefinite.
I could run faster than most people today, if I really wanted to.
The above sentence is in the present subjunctive, not the past active. It provides the sense of uncertainty arising from 'if', to say "I am able to run faster if I want, but whether I do or not is not currently certain."
So your suggestions for what 'could' means are actually exactly right.
1) I could take her to one of the group homes.
Means "I am able to take her to a group home, but I have not yet received clear instructions to do so. I have the ability, but it is not clear yet if I need to."
2) Can you use 'might'? Yes.
'Might' is another way of indicating uncertainty. 'Could' has more of a sense of "I am able to; I can". So it indicates ability.
I 'might' do something is most commonly used to say "I can do something, I have the ability, but I have not yet decided whether I will. I'll do it if I want to, but I have not made a decision.
So it indicates a state of indecision by the subject of the sentence.
I might take her to one of the group homes,
Means "I am capable of taking her to one of the group homes (I can take her), but I have not decided whether I will take her.*"
In modern informal English, there is a strong preference for 'can', 'could' when talking of possibilities. Some people consider this grammatically wrong, but in 'real life', it is very common.
Can you open the door?
The above sentence is often used to ask someone to open a door. In other words 'would you please open the door for me?' In a strict logical sense 'can you' means 'are you able to/do you have the ability to', to which the answer for almost everybody is 'yes'. Everybody can open a door. Strict grammarians would suggest we all should say:
Would you open the door?
We know you can open the door, but will you please do so, right now?
In real life, the differentiation between 'can'/'could' and 'will/'would' is very rarely observed. 'Can/could' are used a lot more often than 'will/would'.
But ... if you want to sound really smart and expert at English, say 'will/would'. To many people it will sound 'fancy' and 'clever'. So long as you are not really asking whether someone can do something.