Yes, phoenix does count as a mythical creature, but it cannot use nor control Greek fire (which is a weapon not just fire) - depending on sources it's can be normal fire or other kinds of mystical fire (certainly
not Greek fire). Again, I have to repeat this that since Celuis isn't related to Hephaestus in any way, she should not have any power over any kind of fire.
Even for Hades he doesn't have any power over fire - earth and metal yes,
not fire.
You might want to read up for more details about the world of Percy Jackson and Greek Mythology as depicted in the book series on the
wikia. Hope it helps you to better understand the lore.
EDIT: Eh like I already told you before, I cannot keep up with speed-posting. See what happens when you do that? I end up repeating myself in my replies... /sweatdrops