Ha, I haven't been back to this forum for months (been a bit consumed by work), but I loved reading this post! I'm in almost exactly the same situation, 4 kids between the ages of 2.5 and 7, of which the elder three girls are fascinated with daddies books filled with dragons, swords and elves! As I started aged 7 (back in 1979) I thought I'd give them a try to see how they liked it. I have all editions except 4E, which felt like a huge betrayal to me, yet deliberated for a good 2 minutes as which to use. The answer was all of them! They're too young to really understand the mechanics fully so I use the 1st edition to hit tables as the basis but the rest is largely fudged. AC to them is qualitative, leather is "better than nothing", plate is "really hard to hit". Weapons are "huge" swords, "nimble like light-sabres", "fists like iron" etc. One daughter wanted to be a Magic-User and Cleric but couldn't understand why she had to divide her HP's - viola quick 3E adaptation fixed the issue. The result is they are loving the story-based adventure I'm running and not getting bogged down with what I'm doing behind the screen.
As for which module/ adventure to use - it had to be the original B2 Keep on the Borderlands! Very easy to adapt from basic rules to my home-blend, with simple tasks and concepts for their age. For alignment, one of my daughters has just written "goodies" - that's fine for our game. Perhaps over the coming years they'll become rules lawyers and I'll be found out, but by then I hope they'll love the game and my job will be done!
If your kids are slightly older I think A1-4 are fun, with some moral issues to wrestle with regarding slavery. As mentioned I3-5 are great because the Egyptian theme is something they can probably relate a bit to through school. T1-4, or the GDQ series are a bit heavy on symbolism and played properly should get really 'evil' so I'd avoid these unless your kids are a lot older and mature.
Anyway, good luck, I hope it goes well and keep us updated!
Cheers,
Vaprak