All of the greek and roman myths (they're not considered myths within the story) include gods and other characters marrying their siblings, children, etc, same as any story containing greek and roman myths. For the most part the story skirts around it.
A lot of battle scenes. The ones with monsters can be a bit gruesome, but monsters turn to dust when they die, so the gore isn't as intense as it would be for mortals.
Leo has the gift of fire, and occasionally has controlled flames run across his body. They do not hurt him at all. Frank's life is tied to a piece of burnt wood. If the wood burns all the way, he'll die, so this is referenced occasionally. A giant gets blasted with Greek fire.
A group of "american tourists" are described by Leo in very unkind terms. They're later revealed to be not human, but the sentiments are harsh and rude, and go on for a little while.
No, but Percy has a dream in which Gaia gives him a vision of what she will do to the earth if she succeeds, and it's a lot like the destruction from a nuclear explosion. It's pretty awful.