Could a free sprite or AI (which are both considered Resonance beings) make a home node out of a Technomancers living persona? If so what would happen if the Technomancer was against the idea? Or is this all some way for me to justify my schizophrenia?

I'm pretty sure that an AI is not a Resonance being and is therefore unable to enter a TM's biological node.

I don't have Unwired handy, so I can't really reread the section about the Living Persona to answer that part of the question. However, I will say that if a PC wanted a conflict like that I'd be hesitant as it would lead to non-stop potentially lethal conflict. On the other hand, if the rules support this type of thing then it might make for the backdrop of a fun run.

No it's not possible for an AI to access an Living Node since it isn't a resonance being and therefore it can't set up it's home node there. Plus, it couldn't even load itself onto it since a Living Node has no storage memory.
And a Free Sprite doesn't need a Home Node.