The Question and Answer section for The Lightning Thief is a great resource to ask questions, find answers, and discuss the novel.
Percy's cabin belonged to Poseidon: it was definitely ocean themed. Water can also heal Percy and at one point a glowing trident appears over his head.
Annabeth used to have a dog and plays ball with Cerberus. This seems to tame Cerberus.
I think that Rick Riordan justifies his story by claiming that is is rooted in ancient Greek history which means mythology but we will call it history because his story is so good!.