Learn English – If you came [had come] tomorrow, I might be [have been] able to help you.​

A very thoughtful and hard question indeed; it pushed me to research a bit on the subject Continuous Conditional.

Your first example is a past perfect continuous sentence. In general, it is used to indicate an action was happening before another action happened. However, it can also be used to indicate past unreal condition. For example:

If I had been talking to him when he said that, I would have punched him in the face.

But fortunately, he was not talking to him when he said that and that's how he missed my punch.

So according to this theory, your first sentence which is

If I had been with you, I might be taking care.

is absolutely meaningful and grammatical.

Now, there is a vital fact when using these type of constructs as described by data.grammarbook.com,

When talking about something that didn’t happen in the past, many English speakers use the conditional perfect (if I would have done) when they should be using the past perfect (if I had done).

For example, you find out that your brother saw a movie yesterday. You would have liked to see it too, but you hadn’t known he was going. To express this, you can use an if – then clause. The correct way to say this is with the past perfect in the “if” clause, and the conditional perfect in the “then” clause:

Correct: If I had known that you were going to the movies, [then] I would have gone too.

The conditional perfect can only go in the “then” clause — it is grammatically incorrect to use the conditional perfect in the “if” clause:

Incorrect: If I would have known that you were going to the movies, I would have gone too.

More examples:

Correct: If I had gotten paid, we could have traveled together.

Incorrect: If I would have gotten paid, we could have traveled together.

Correct: If you had asked me, I could have helped you.

Incorrect: If you would have asked me, I could have helped you.

The same mistake occurs with the verb “wish.” You can’t use the conditional perfect when wishing something had happened; you again need the past perfect.

Correct: I wish I had known.

Incorrect: I wish I would have known.

Correct: I wish you had told me.

Incorrect: I wish you would have told me.

Correct: We wish they had been honest.

Incorrect: We wish they would have been honest.

So this theory, in a nutshell, says you can't use "would" part with the "if" clause, rather it should be used in the result clause. So, according to this theory, your second sentence stands incorrect.

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."