This does make sense and I think that the bombs aren't filled with mutagens or something, the changes are just part of regular nuclear reactions maybe mixed with magic, but I think the bombs themselves are definitely just regular nukes.
Why would Marceline lie about what's in the bomb?
She doesn't even have to have been a demon on FarmWorld. Who's to say that Simon, who showed control over the crown, did not extend her natural lifespan in some way? The crown gives an extended lifespan to at least its wearer so it is in the realm of possibilities.
She is definitely half-demon. Her dad is still Hunson Abadeer. Hunson Abadeer is a demon. Ergo, she is half-demon. Simple.
Maybe the War was a war of extermination between humans and Ice King!Simon/Magical creatures. We know that magic was coming back before/during the War as Simon, Marceline and the Lich are from that age. Human history shows that plenty of wars where started over the simply reason of "It's either us or them.".
I doubt that. The only known supernatural beings before the Mushroom War are Simon and Marceline, the Lich was a product of the Mushroom War. If the bomb was dropped on Simon specifically then we would have seen him facing off with the planes in a frozen waste-land rather than on a grassy hill. We can assume it was simply a world war scenario where one side went too far.
We don't know how old Finn is. FarmWorld!Finn was trying to save people by freezing them. We know that people who are frozen can survive the unfreezing-process. (See the Businessmen). Maybe they are pulling an "Avatar:TLA" and Finn's parents died during the Mushroom War while Finn himself survived due to being flashfrozen. Correct me if I was wrong but Jake's parents found Finn wandering in the woods. They never met or saw Finn's biological parents.
1. We know how old Finn is, he started out as 12 in series one and now he's 14. He'll probably be 15 by the end of this season.
2. You're right that Jake's parents never met Finn's, but the rest of that ain't right yo. For one thing, FarmWorld is set in the same time frame as Ooo, therefore Finn's parents must have been born after the mushroom war. I doubt we'll ever learn about Finn's actual parents beyond what we saw of them in FarmWorld.