Ami's aware of his tutoring:
Even remembers him doing it more back when she was in elementary school.
She just can't remember Noriko, and definitely not Maya. Since apparently Sensei didn't meet Maya until Ami was in middle school:
In other words, she knows he tutored. She just didn't know he was hiding stuff at that time like his relationship with Maya.
Ami also seems to remember living in the old district:
so if Sensei had lived at the ramen shop building with Tsuneyo etc, then I think Ami would know Tsuneyo.
From what I can put together, Sensei was living in his apartment when Sekai died. Then he took in Ami, and lived at the apartment until they got better then moved into the current house. How he snuck his tutoring around Ami is beyond me, unless they were living at two homes simultaneously, and he merely used his apartment for tutoring and never told the Nakayama's about his current house.
What makes you say the god of wires doesn't want Sensei to regain his memories? Perhaps I missed something.
As for Tsuneyo's skin color, it's not that it's weird. It's just that when she mentioned playing with some girl earlier on that could have been Noriko, she only mentioned the girl had different hair. If it was Tsuneyo, I'm not sure why she wouldn't say "and had darker skin".
Yuki: my bad. I tend to take things far too literally.