Ending of Shutter Island

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.

As far as I know, if you commit some crime and are found mentally ill, you do not go to jail. Instead you go to a mental hospital, where you are treated until you are fine. Then you are free. Hence the whole "pleading insanity" option in various movies/tv shows. It would not make sense to stay 3 years in a hospital and then also get the full sentence.

This being said, lucidity would not also bring his arrest.

Now, as to the whole problem, to my mind he is cured at the end, but keeps pretending in order to be lobotomized. He realizes that he cannot live knowing the truth, preferring this solution instead. The doctor's nod means(to my mind) that he gives the permission for lobotomy. It is a sad look, so it cannot mean that he tells them that it worked.

Here is a similar question and the accepted answer provides more details.

There still remains an unanswered question(nobody has asked it, but I believe that there is also no proper answer to it): Does the doctor understand what Teddy means? More precise, is aware that Teddy is cured and approves the lobotomy as a favor for him(the gift of mercy) or is he simply fooled by Teddy's acting and does not understand the meaning of the last sentence?

Dr. Cawley was attempting the role play exercise in an attempt to bring Andrew back to reality. The experimental treatment was an alternative to the conventional treatment of lobotomy. Cawley indicated that if the role play treatment did not work on this final attempt, he would have no choice but to follow a standard treatment. At the end of the scene, Andrew admits his crime and appears to accept reality.

In the following scene, we see Andrew appear to relapse. I believe the significance of Sheean's gesture is that the treatment fails and that a lobotomy will now be administered. Andrew goes willingly.

It is my opinion that Andrew is aware (now, if not always) and chooses this fate rather than live in the knowledge of his crime. Speculation: As a man of violence, he would rather accept the 'violent' procedure than back down and face defeat. He stays consistent with his character and lives on his own terms.

was Teddy really cured?

No, in a experimental procedure the psychologist let him play out his fantasy in order to reveal the truth about his crime, but he understands it for the brief moment and comes back to square 1 the next day.

the significance of Dr. Cawley's eye gesture at the final scene?

Like i said the procedure was experimental and it failed the next day (maybe because of his mental condition) so the assistant[who pretended to be the partner] shares a smoke with him to confirm if he is able to come in terms with reality and makes a gesture to the doctor saying "needs more time".

I think the scene where Leonardo DiCaprio is put in the psycho's place - where he's killing his wife - is just him contemplating what the people are telling him. The idea, along with all the frustrations and horrible danger he's in, makes him faint. I think everything about Teddy is real from the beginning, and the owners of the asylum are villains. At the end, Teddy makes a comment about being too smart, and it's implied that he has a plan when he looks at the lighthouse.