I love the idea! The Silmarillion is an yearly read of mine either way so I hope you won't mind me joining.
It always fills my heart with joy when others read it and therefore it might be a better idea to wait a bit. But I'm on board either way.

However there are a couple of things I'd like to "set straight" though (hope I'm not sounding too dickish here)

Eru, or Ilúvatar sang songs with the Ainur

Not quite. Initially only the Ainur sang for him. Later he declared to them themes but it was not until the discord of Melkor that he sang alongside with them.
It is just a minor detail, but if we get into "theological" debates it makes a bit of a difference.

"Melkor" who was favored in the eyes of Ilúvatar, and who had the greatest gift of power and knowledge.

Again, not quite true. He is the mightiest and most gifted, but it is said that Manwe was dearest to Eru and that he knew most of Erus mind.
Namo (Mandos) and Manwe certainly know more than him, yet Melkor is the most gifted and "mightiest", though Tulkas is "stronger" and Aule is only a tiny bit behind him in craftsmanship.
Also, Melkor is an Ainu, Ainur is the plural term. Just like Vala is the singular of Valar.

Ilúvatar though, then only changed the melody to include Melkor's discordance
I'd use the term subvert. It was already part of the music, he just changed the music around it.

Ilúvatar showed the Ainur a vision of Arda - what they had then created.
Technically what they had to yet create/form. He actually created the essence world but left it to them to form it.

They looked into the mind of Ilúvatar
I disagree. He showed them a vision in the void and did not reveal his mind or let them enter it.
As for Melkor's shame and anger one might want to add the "speech" of Eru in which he told Melkor that in the end, even though he may not realize it, his "rebellion" is part of the plan and will only contribute to the greater glory of the creation.

The children being Elves, Hobbits, and Men.
Hobbits are part of men and are nowhere fortold. They come into existence much later.

...but through that they also began to see the things that Melkor had brought, most important being darkness.
I cannot remember this part at all, I don't think it was explicitly mentioned.

The main laborers at this time were Manwë, Aulë, and Ulmo...
Hey! Don't you forget Varda and Yavanna! :-P They played huge parts. Varda's stars, especially, are the only thing that Melkor had not corrupted or interfered with and he was most afraid of her and her light if I remember correctly. And Yavanna just happend to create all plant/animal life.

called forth many spirits to help aid to hinder Melkor
Technically not quite true again. Many had gone with them initially to serve and help (being of lesser power); the Maiar. Some of them were corrupted by Melkor, who would destroy and pervert anything they created and himself created mostly great fires. He didn't flee until Tulkas came to aid the Valar in their fight against him.

The Valar then decided they would like to take form, close to what they had seen of the Children of Ilúvatar.
Ulmo for example didn't. And especially in the beginning they are told to have taken many shapes. I don't think that they "settled" for the "elf-form" until their first encounter with the Quendi, but that is open for debate, given that I don't remember it explicitly mentioned.

Thus began the first battle of the Valar with Melkor for the dominion of Arda
I disagree,it had been going on. And he wasn't able to fully destroy it, he only changed it in an attempt, which is the reason why the world isn't "perfect" as planned or symmetrical at least.