1. Yes, English is a bit less forgiving of verbs with deferred subject than Spanish is, so it can't be done as cleanly as in Spanish. But it can be one better that I did, instead of "that she who has gone away should come back to me" the last line would be better written "that she should come back to me, she who has gone away". But your number 3 makes me think it should be simplified a bit more, see below.

2. I don't like "left" here, it feels like the wrong verb. "went away", "has gone away", "is gone" all feel more natural. Using "come back" and "go away" as opposites. But that's just my personal feeling, perhaps some people would find "left" perfectly ok here. And "left" wouldn't seem such a misfit if "return" was used instead of "come back", but I prefer "come back" in this song.

3. Originally English would have had a subjunctive here, "that she come back" would have been the most recent form of that but that now sounds rather archaic. Instead of the subjunctive people use all sorts of constructions with auxiliaries (should, would, could depending on context and personal preference) or be with infinitive ("were to come", "was to come", "be to come") not usable in the context of these lines, but very common in some contexts. Another possibility is to avoid the subjunctive by finding some means of avoiding a finite verb altogether, and that may be the best way to translate these two lines, because it is far more common usage to express desires than using a clause introduced by "that" is: instead of
"Me, what I want is that she should come back,
that she who has gone away should come back to me."
it could be
"Me, what I want is for her to come back,
for her to come back to me, she who went away."