• The Goddess Artemis, Demeter and other nature gods have been collaborating in protecting pockets of nature and assigning nymphs or her Hunters that for one reason or another do not want to gallivant around the country hunting monsters, through blessings to take care of them, covering them in big layers of mist to keep away the mortals. That is how the haunted forests rumours came about apparently. These Nymphs are strong enough to deal with weak Monsters with their chlorokinesis, even if they are generally peaceful and powerful monsters generally stay away from these places lest the incur godly wrath.
• Natural Dungeons (this is not a gamers fiction) exist scattered everywhere on the world. They are created either by Divine will, magical mishap, or other supernatural shenanigans. God's place different monsters and puzzles in them that you will have to solve. Gods started this, to train their children by giving them a taste of battle in a somewhat controlled environment and for their own personal entertainment. When cleared they always give some type of reward depending on their grade.
Dungeon grade is discerned through colours:
Green- Common
Blue- Uncommon
Yellow- Rare
Orange- Heroic
Red- Legendary
Gold- Mythical
Black (Run motherfucker, RUN!) - Divine
• A god, when taking a champion, normally gives his champion a small part of his domains. Something like a pseudo adoption really, enabling him to tap into them, though not at a level of a true demigod child initially. The connection can of course grow when you bring honour to the name of your patron as a reward, but it is extremely difficult and it becomes more so after every growth.
How big a part depends on the god. Since it is Hestia here and Percy is her first champion and her only way to enforce her will in mortal world, she gave a big part, that normally a demigod children would naturally have. Percy's feats will be great enough to make take them from a normal child to a strong child.
It reaches a point that you need to do something truly spectacular like say kill Ladon solo to broaden the connection. In case of Percy, it is not a problem since he will be killing gods and titans every year.
• The Mist is more than just a veil to hide the mortal from the immortal word. This faction was introduced much later after the fall of Rome. The Mist is the ambient mana of the world that mix with all the supernatural entities magical emissions absorbing the special reality bending energies the Gods release just by existing. The sheer capabilities one can achieve by learning how to use it is truly awe-inspiring. Illusions, Astral, Holographic, and even material Projection, territory creation, dream manipulation and many others countless uses that it would take a whole day to tell them all. It is basically all the esoteric branches of magic in one badass package.
• All gods have many helpers in form of there demigod children, mortals, that they have given partial immortality in return for them becoming their 'priest', sort of. They help the god's in spreading their domain and work as their personal army, just like Hunters of Artemis.
• Deities like Poseidon who have their own physical domain only keep half or even less than half of their power in their Throne on Olympus. The other part is in Atlantis. Same for Apollo and Artemis who have another Throne on the literal Sun and Moon and compete with other sun and moon gods for territory and thus power. Also destroying thrones does not mean a permanent loss in power, if the Throne is destroyed but the god manages to hide, for like a month than he will be able to regain the energy back. But the temporary depowering is enough that if you are in a conflict at that time, like the Battle of Olympus, than they will surely lose. The energy is too much to carry in your body all the time that is why the Thrones are made.)
• I am also adding another thing I thought about. Every element has a spirit of it in Greek mythology other than fire like-
Water - Naiads, Nereids, Oceanids
Air - Aurae and Harpies
Earth - Dryads and various fruit and plant spirits in Trials of Apollo
Sky - Venti
So, I have added fire Nymphs as well, they are the Nymphs of fire, like forest fire are the wild ones, most are those of your hearth and campfire, the fire on which you cook food, the fire you use to protect yourself etc. They are loyal to Hestia and Hephaestus other than a few rogue ones who are loyal to Titans like Hyperion, Perses etc.