Are there any parallels between the Faith of the Seven and Lovecraft? If so, are there any deities in the Lovecraft mythos that each of the Seven might correspond to?

The Faith aren’t really Lovecraftian. They’re pretty straightforward Fantasy Catholicism. 

There are Lovecraftian religions in ASOIAF: 

• the Faith of the Drowned God has a lot of Cthuloid undertones.
• In ADWD you have the Cult of Starry Wisdom, which is a straight lift from Lovecraft. In “The Haunter of the Dark,” they’re a cult of Nyarlathotep worshippers based in Providence, Rhode Island.
• The Black Goat of Qohor is pretty close to Lovecraft’s “Black Goat of the Woods with a Thousand Young” which shows up in “The Whisperer In Darkness.”