Does the aforementioned divorce constitute a Khula' or a (regular) divorce? And is the wife obligated to observe the 'Iddah if consummation did not occur but privacy did? And is it permissible for her to get engaged to another man?

If your husband divorces you and you do not agree on a compensation for the divorce, then this is not a khul‘ (divorce at the wife's instance with compensation) and he is not entitled to compensation. If he gifted you something for the completion of the marriage and it was not completed, he may retract his gift. You are entitled to half of the mahr (dowry), and there is no ‘iddah (waiting period) for you if consummation or valid seclusion did not occur, and kissing and the like are not considered consummation. You are permitted to accept new marriage proposals.