Claimed status means 2 things.
1. If you are claimed, you live with your siblings in your parent's cabin. Aka Apollo's Cabin.
2. It also means you sit with your siblings during meals.
3. If you are NOT claimed, you sleep and eat with the other unclaimed children in Hermes's cabin ( the largest and most populated, as it also houses children of Hermes)
4. Being claimed also means you have proven yourself as a child of that deity. Being claimed can range from proving yourself a great combatant (would work more for Ares kids), tactician (Athena), Crafter (Hepatsus), etc etc etc.
Fatal flaw= Are your weaknesses, what can be your downfall. Normally good in small dosages but to much can be your ruin.
Example- Franky is bullheaded. So even if there is reason behind NOT doing something, he is so stubborn he will do it to be right at the risk of his own life.
These are important and SECRET, cuz once someone knows your fatal flaw, it is basically your demise.