Lea was at the Battle of the Stone Table. Mab mentions her to Harry. She was with Maeve. Maeve says to Harry, "[y]our godmother sends her greetings."
Aurora refers to a godmother as a teacher and a guide. She says Lea taught Harry "[t]he strong conquer and the weak are conquered" after Harry says the only thing Lea taught him was that he must be smarter and stronger than Lea and willing use those advantages against her.
Lea confirms that Harry was her student in
Ghost Story. She says teaching "is the dissemination of power unto a new generation and nothing less." Having the Fomor attack Molly was a lesson to Harry as much as it was to Molly.
After Lea delivers her monologue about how she has always protected him with her garden and Harry realizes that was why she was always quick to attack him in the Nevernever, because she was always there protecting him from everything but her, she says:
It is good that you have taken my lessons to heart, child. It is a cold and uncaring universe we live in. Only with strength of body and mind can you hope to control your own fate. Be wary of everyone. Even your own protector.
I think all this (except the first part, which is just something I noticed looking for Aurora's quote, but bears on the last point in the original post) supports my contention that Lea warned Harry away from the Stone Table because she was his godmother and not because she was infected.
The first part may have been when Lea spread the infection to Maeve.
As to what Lea being Harry's godmother entails, we know it entails teaching him and protecting him, specifically from threats to his home directly from the Nevernever. It could be more.
One question this leaves for me is how did the Nightmare get to him in
Grave Peril? It probably has to do with Harry traveling to the Nevernever in his dream, but I'm not sure.