Could have is saying that something was definitely possible in given circumstances.
Could have indicates that the option was available: that the possibility existed, not limited by circumstances, situation or personal ability. It tends to be used when you are talking about an entire hypothetical scenario.
When I was training for the marathon, I could easily have beaten you in a race!
The focus is on the capability: in this case the speaker knows that they definitely could have won the hypothetical race if such a race had occurred.
Note that saying they are definitely capable of doing so doesn't necessarily mean that it would definitely have happened, just that it had a probability of more than zero. A negative example might help to illustrate this:
Even if I had trained for a year, I could not have beaten Jesse Owens in a race.
It also doesn't give any indication of how likely the speaker thinks the possibility was, just that they believe it was definitely a possible outcome.
Would have defines a definite outcome given different circumstances.
Would have indicates that something intervened to stop the predicted outcome from arising. It tends to be used when you are talking about a change to the real sequence of events: if something had happened differently.
I would have beaten you in that race if I hadn't tripped up!
This is expressing certainty: if the situation was different (in this case, not tripping up), the predicted outcome (beating you) would definitely have occurred.
Might have defines a possible outcome given different circumstances.
Might have is the same as would have, but the speaker is less certain of the outcome.
I might have won that race if I had done more training.
I didn't win the race, I think I might have won if I'd done more training, but I also think that I still might not have won even if I had done more training.
Your specific examples
As you might have guessed, many of these sentences are correct but with subtle differences in nuance.
Example B
- If I hadn't been so tired, I might have realised what was happening.
- If I hadn't been so tired, I could have realised what was happening.
Because this is talking about a change from the actual situation, not an entire hypothetical scenario, "might" sounds more natural here than "could". That doesn't mean that "could" is wrong, just that it would be more common to say "might".
If you were sure that you would have realised what was happening if you'd been less tired, you could say "would" instead of "might".
Example E
- Why did you jump out of the window? You could have hurt yourself.
This is indicating that hurting yourself was definitely an option (and hence it was stupid to jump out of the window). For this reason, it expresses the sentiment most strongly out of all your "window" sentences. It's implied that you were lucky not to hurt yourself - in other words, luck was all that prevented you from hurting yourself.
- Why did you jump out of the window? You might have hurt yourself.
"I suppose you might have hurt yourself, but I'm not a window-jumping expert, so I don't know..." There's less certainty here, so it's less forceful. However, it's still entirely acceptable.
- Why did you jump out of the window? You would possibly have hurt yourself.
"Would possibly have" has the same meaning as "might have". To me, it sounds slightly less natural here, but I think that's because the question itself sounds a little accusatory, so adding the "possibly" in is a little awkward.
Example D
You didn't suggest a substitution for D but I'm including it to show the difference from E.
- I was so angry I could have killed her!
- I was so angry I might have killed her!
In this instance, it's actually the "might" example that is stronger. That's because the "could have" suggests that the speaker was angry enough to kill her, but chose not to. Because it's "could", we know it's a hypothetical scenario and that there was no chance of him actually killing her: just that his anger gave him the capability of doing so.
However, "might have" admits the possibility that this outcome might actually have occurred. It sounds like it's something external to the speaker that prevented it, like someone walking in, or maybe he went so far as to beat or shake her and it was only luck she didn't die.
Needless to say, "would have" is therefore the strongest of the three, indicating that she definitely would have died if something had been different (like they didn't get interrupted).
Example F
- I could have won the race if I hadn't fallen.
- I would possibly have won the race if I hadn't fallen.
These both sound equally natural, but the nuance is different. In the first one, the speaker is certain of their ability to win races, but acknowledges that that doesn't mean they'll win every race they ever run. Falling stopped them being able to win the race - an ability they had before they fell.
In the second sentence, on the other hand, the speaker is still acknowledging that the outcome of the race (even without falling) was uncertain, but they are doing it in a way that includes the possibility they might not have won the race because they simply weren't good enough. It's a broader statement: that they might have gone on to win if they hadn't fallen, but they don't know if they would have gone on to win.
Some other things you mentioned in comments
I think your would possibly/perhaps have example is the same as might have, and would have been able to is the same as could have, with would have been allowed to being a subset of that where permission is the limiting factor of the "could".
Tom is the best expert ...
Assertion of a fact. Where there are agreed criteria then there's no need to hedge an statement.
Djokavic is the current number one male tennis player in the world
When criteria are less certain, or we ourselves are not sure of our facts, or we wish to be modest by appearing to be uncertain of our facts we may use I think
I think Djokavic is the best male tennis played of all time
I think that the population of the UK is 50 million (actually 65 million)
This formulation implies that we are open to correction and discussion
Adding would softens this further, emphasises that we are uncertain
I would think that the population of the UK is greater than 50 million
The I would have thought formulation is normally used in a context where some information has recently been given. Depending upon the context it may imply that we are actually contradicting the information, or that we are expressing surprised acceptance.
I think Tendulakar is the best batsman of all time
I would have thought Bradman had a better record
that was disagreeing, suggesting that by some criteria Bradman is better
The current UK population is 65 million
Oh, I would have thought it was only 50 million
but now I've changed my opinion (this implied but not said)
that was agreeing, I thought it was 50 million, but I accept your statement of 60 million is correct. We could just say
Oh, I thought it was only 50 million
With pretty much the same meaning, the slight difference being that the second case implies it was actively in my mind, whereas the would form could imply that I hadn't really formed a solid opinion until now, but I would probably have guessed 50 million.
As your comment indicates we are indeed into shades of meaning and idioms. In these cases the tone of voice will often differentiate the meaning.
I would have thought Bradman had a better record
Would probably said with a questioning or challenging tone and raised eyebrow.
Best Answer
There is little or no difference in meaning between the two sentences. There is a tiny difference in emphasis, which is hard to explain. It has to do with the way you clarify or remove ambiguity in English by adding redundancy, also illustrated here.
Hypothetical possibility with present tense
In both sentences, "his doing my homework" is a hypothetical possibility. A consequence of that possibility, indicated by would, is that the speaker would watch a film.
English provides a variety of ways to indicate a hypothetical possibility. The simplest is just to precede the clause describing the possibility with the word if. You could say:
However, in this form, the meaning of the present tense is not clear. Does it mean "right now"? No, that's not how the present tense is normally used in English. Does it mean "in general"—that is, "if he often, frequently, usually does my homework…" or "if it is his responsibility to do my homework…"? This is how the present tense is most commonly used in English, but it's probably not the intended meaning in this sentence, since "I would watch a film" suggests that the speaker is talking about only one occasion. However, would can also mean a habitual, recurring action. The little English grammatical words tend to be very ambiguous!
So, in English, we often add clarity by repeating or echoing the same meaning in multiple ways. This helps reduce the ambiguity of each individual word or grammatical choice, especially regarding what the speaker means about the time of an event, whether the event is real or hypothetical, what the consequence is and when it occurs, whether the hypothesis is normal or unusual or an offer in a negotiation, etc.
Past tense to indicate future possibility
To avoid the ambiguity described above, it's common in English to describe a hypothetical future possibility by putting it into the past tense, like this:
Another typical example:
You might think that using the past tense to indicate a hypothetical future possibility is ripe for confusion, and you would be right. For example, "If he did my homework, I would watch a film" could also mean a past possibility and a present consequence: if he already finished doing my homework, then I would watch a film right now.
So, when using the past tense to indicate a future possibility, people often add additional words to reinforce the interpretation that the sentence is about the future. For example, the sentence with "visited" has the adverbial phrase "next July", which makes fully clear that that the sentence is about a future possibility.
Subjunctive mood
The sentence about homework does not have an adverbial phrase to make the time explicit. So, to be perfectly clear that you are using the past tense for a future possibility, you might use the subjunctive mood. The subjunctive mood means hypothetical possibility! The subjunctive mood is rare in English, and often the form of the verb doesn't unambiguously indicate the subjunctive mood, but with the verb "to be" and subject "I", the subjunctive mood is completely unambiguous, since "were" can't make grammatical sense in any other interpretation:
Infinitive verb form
The reason for the infinitive to do is because the infinitive form of a verb avoids attaching it to a particular time. (For more about that, see this answer.) The past subjunctive were, combined with if, adds somewhat more redundancy to indicate that the sentence is talking about a hypothetical future possibility and its consequence.
Conclusion
Because the sentence talks about homework and watching a film, it sounds like the speaker is referring to hypothetical events that would occur this evening. So, you don't really need the subjunctive mood to be clear about what time is being talked about. The sentence with did is clear enough.
Using the infinitive to do instead of the finite did creates a slightly more abstracted feeling about the possibility. Using the past subjunctive were where it's not necessary adds to that feeling of abstract possibility. As I said, the difference in meaning is just a tiny shade of emphasis. But you can learn a lot about English grammar by understanding why the switch from did to were to do creates that tiny shade of difference!