Lost opportunities
The meanings of these are indistinguishable, except for a subtlety that I'll explain later:
We may have been able to save the victims.
We might have been able to save the victims.
The words "would" and "could" suggest both the past tense and the consequence of some imagined hypothesis, with either or both meanings being activated when suitable to the context. To use "would" here, you need to establish an imagined hypothesis. Then "would" indicates the consequence:
If only we had the antidote yesterday! We would have been able to save the victims.
It's possible to say "could have been able to", but "could" is the past tense of "can", so it already indicates ability. So, "could have been able to save the victims" is awkward. However, you could say:
If only we had the antidote yesterday! We could have saved the victims.
"Would have" and "could have" imply that the opportunity is now lost.
"May have", "might have", and "could have" don't require context to set up an imaginary hypothesis, but they do suggest that you have such a hypothesis in mind:
"We might have been able to save the victims."
"How?"
"The most recent antidote has been sitting in the refrigerator since yesterday."
"What? And no one tested it?"
An important difference
A very big difference between the may/might and would/could sentences is that the would/could sentences claim a high level of certainty about the hypothetical situation: the antidote is assumed to work. The may/might sentences lack that certainty. They suggest only that in the hypothetical situation (which might not even be stated yet), the antidote might have worked. "Would" corresponds to "will", and says that the antidote will work (in the imagined situation, which occurred yesterday). "Could" corresponds to "can", and suggests that the antidote can work (in the imagined situation, which occurred yesterday).
A subtle difference
"May have" mainly indicates uncertainty about what we could have done. "Might have" suggests both uncertainty and that the opportunity, if it even existed, is now lost.
This is English, so don't take that as a rule. Instead, memorize this well-known couplet (by John Greenleaf Whittier, 1856):
For of all sad words of tongue or pen,
The saddest are these: "It might have been!"
One more power of might
There's one more possibility with "might", which illustrates the flexibility of these words. "Might" can suggest that the victims actually were saved, or are actually in the process of being saved:
We might have been able to save the victims. I just read an article in a medical journal, which says that the antidote we tried yesterday was proven to work on mice in a recent experiment.
In other words, the antidote we tried yesterday might be working, right now. Since the uncertainty of the word "might" applies to our ability to save the victims, if we had that ability yesterday, then our efforts yesterday were successful, or are on the way to succeeding. Currently, we don't know (that's the uncertainty of "might"). Obviously, your listener will hear this meaning only if the victims are still alive.
You can also do this with "may have" and "could have", but those don't make the pun in the title of this section. Since "may have" has the least connotation of lost opportunity, it provides the clearest way to express optimistic uncertainty about the effects of yesterday's action: "We may have saved the victims."
That's just the structure of the exercise. Where the blank is, it isn't grammatically correct to say "could". But if the exercise allowed you to change the sentences a bit, you could very well use could.
In the first sentence, the correct syntax is used to + verbal root, so you have to put be able to. You wouldn't say:
26.1.4 I used to could stand on my head, but I can't do it now
But you could say:
26.1.4 When I was young, I could stand on my head, but I can't do it now.
Which is grammatically correct.
Same story for the second one:
26.1.5 I can't understand Martin. I've never could understand him.
Isn't grammatically correct. I've never requires a past participle after that. If you had the ability to modify the sentence, you could say:
26.1.5 I can't understand Martin. Actually, I never could understand him.
Best Answer
I could stay with her, but I didn't.
This means I had the chance but I chose not to or had other things to do.
I could have stayed with her, but I hadn't.
This means I wanted to stay but there was not a chance for it.
These are similar to the construction of conditionals in English. I advise you to see them.
I could have gone to Oxford University but I had preferred Harvard.
I would use past perfect for consistency with the rule.
Here's a reference. https://www.englishgrammarsecrets.com/couldhave/menu.php