The Я can be omitted? Is Russian like Spanish in that personal pronouns can be omitted?
When expressing opinion ("I think that <SOMETHING>"), я can indeed be omitted. Мы less so. Ты/вы can also be omitted in questions about the listener's opinion.
"Я думаю, что она читает." was marked wrong. Even if unnecessary, is putting "что" actually wrong here?
No, it isn't. The sentence makes sense with "что" too. Added it.
was this decision reversed?
"думаю что она читает" was rejected (as of 3/18/2019)
the correct response was given as "Думаю, она читает"
I put that and was marked wrong too.