"Elemental Hero Divine Neos" will have its name treated as "Elemntal Hero Noes", but maintains all of its other effects. When you second summon a Gemini Monster, since it does not leave the field, any previously placed conditions or effects stay "attached" to the Gemini Monster. As such, Divine Neos, after gaining Alius' effect and being "second summoned" other than having its name changed, still has all of its other effects. This lasts, as Divine Neos' text dictates, until the end of the turn.
Sorry this took so long to answer, I guess I missed yesterday's session here (I check in once a day).