I just realize how close that name resemble, but Yggdrasil pronounced "ig-druh-sil" so maybe still not the best match.
Both are correct, Yggdrasil is supposed to be pronounced that way, but in Japan they romanized it as yugudora(shiru) when they don't just call it world tree. At the very least I know games like Dragon Quest and Yggdra Union called it that.
I dunno. Cz Sefi call him "Yugu".
That's just japan's way to shorten names, they usually pick the first or last two letters of the romaji.
Reminds me of another anime where the character's name is Beatrice and one of her friend wants to give her a nickname so she called her Beato from Beatorisu.
tl;dr
I can't say the author intent for sure, but if they do want her to be yggdra that is the expected way they would go about it.