My answer is just the opinion of me, and not based on hard proof. That said, I think Teddy was really cured, but he just pretended that he was still investigating the case. That way, he wouldn't get arrested for killing his wife and could just "role-play" his life again and again without arousing any suspicion.
For your second question, I think Teddy just made his game (pretending he wasn't cured) seem more realistic. His conversation with Chuck before getting up and walking away convinced Chuck that Teddy really is not cured. That may be the reason Chuck looked at the doctors(?) and slowly shakes his head while maintaining eye contact.
Another theory is that (for your 2nd question), Teddy just confessed his cure to Chuck in that conversation before getting up and walking away. But Chuck decided not to tell the doctors(?) of Teddy's cure. 'Cause if he did that, Teddy would definitely be locked away. Teddy's constant role-playing with Chuck may developed some connection (pity? friendship?) between them- and Chuck decided not to give up Teddy.
This is just my opinion and the way I interpreted it and may be wrong.