Thor, Poseidon and Amaterasu appeared to Taylor because they each share a Domain with her
Taylor has Storm and Ocean as two of her non-gifted Domains and Thor is a storm god while Poseidon is a Sea God. There have also been hints that imply Taylor has Sun powers.Alternatively, Taylor will have Star Powers - which will include the Sun as modern Society knows the Sun is a star itself, and as Inari says, perception can affect domains, like how foxes are considered tricksters in Japan, granting Inari shapeshifting and illusions as a result of being the goddess of Foxes.
Part of the reason Armsmaster is so focused on the Trident is because it is so obviously linked to one of the Wards
While Armsmaster is wanting more recognition and fame, he will try to do his duty. A mysterious artifact that pretty screams it has something to do with the new kid on the team has him worried. So to him, the Trident is not only insulting his pride due to its ability to not be figured out, it's also a potential threat to a Ward. And officially, the Wards are his responsibility. One of Taylor's Domains will be Arts
And she will expand that to cover a lot. Music is an art! Cooking is an art! Woodwork is an art! Gardening is an art! Sewing is an Art (and she can make herself clothes instead of needing them custom made via the PRT).