Lord Melkor Rightful Master of the Fates of Arda

[Legate of Amon Lanc]

More or less, yes. The thing I'd point out is:

Or in short, if Morgoth hadn't existed, Eru would have had to create him......

Not actually. If we speak of persons only, then he wouldn't have to create Morgoth - he only would have to create any free-willed person. It could have been an Elf, or a Man (as in the Bible), who would make the first "falling step". The only prerequisite for it would have to be free will to choose even evil. The prerequisite, as we can deduce, Eru gave already to all the Ainur. He said: "Now play some variations in terms of this theme." Melkor was the one who chose to play "outside" of the theme. But even if all Ainur played according to the theme and created a 100% perfect world (mountains where they should be, seas where they should be etc.), the first Elf/Man could still choose evil, and he will be the first "Morgoth". Because they have free will, the possibility existed.

If you want it in plain English: from the moment when a being capable of do good/evil exists, evil (and good in the meaning as you presented it before, as opposite of evil) exists. Singularity (Eru - "the One") does not allow contradictions, because everything is good (!in the meaning I presented here all the time) for it. The second being can cause things the One does not "like" (said very, very vaguely; for using terms for transcendental things is vague at best, but for our purposes let's take it like that, this is not our subject now). As long as the Eä exists, the One can choose how to interact with the Creation (if ever), e.g. in Christianity we have the example that God interacts very much with the Creation, and even is himself the one who completely turns the tide for the Creation by choosing to lead its path from self-destruction (which would result if the Creation were left to itself) to Redemption. In Middle-Earth, the matters are less clear, because Eru intervenes very scarcely - or maybe he does more, but we are not told about it, so this is left to speculation. But what we know for sure even for Middle-Earth is that at one moment, finally, there is the eschatological "Cut!": "Then he raised up both his hands, and in one chord, deeper than the Abyss, higher than the Firmament, piercing as the light of the eye of Ilúvatar, the Music ceased." There is the End of Eä, and after that, there is the Second Music - as I said earlier - and yes, here I'd quote you,
"Hence, at the end, those who participate in the Second Music will sing the themes aright, not because they don't have anything else to sing, but because they will have chosen in full awareness to sing the right themes."