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Arrest
Note that I'm not an actual lawyer, just married to one...
With the separation of Branches of Power, in this case Executive vs. Judicial. Police are not lawyers, and thus they fuck up the law all the time, which is where the lawyers come into play. They can falsely arrest, mistakenly arrest, etc. and can then be punished by the judicial branch later for failure to execute their powers properly.
Miranda Rights only protect what you say while in custody from being used against you (unless they read you your rights, then they can use it). Up until you are in custody, everything you say is fair game, which is why the police were asking you questions before they put you under arrest. Then once they had all the information they needed, they put you under arrest and didn't plan on asking you anymore questions, so they didn't care to read you your Miranda Rights.
Regarding the not being told you're under arrest, at least from the Ohio Bar perspective (assuming it's the same for NYC), you have the right to be told why you are being arrested and the nature of the charges against you (the crime for which you are being arrested). If you are arrested on a warrant, you have the right to see the warrant within a reasonable time after your arrest, to read it and make certain your name appears on it, and to see the charge against you. It's possible that because you never specifically said "why am I being arrested?" (obviously since you thought you were not being arrested), they did not have to tell you why they were actually arresting you.
For them telling you that you're not under arrest, it's a bit of gray area (supposedly police can lie to suspects, as long as the lies don't force the suspect to believe they have no choice but to agree to what the police are saying), but normally it seems that the cops are completely within their rights to lie. Unlike prosecutors and judges, who have professional ethics to comply with, the police are allowed to use whatever lawful tools they have in order to solve a crime.
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