Is my current marriage valid and permissible (halal), given that I married two months after divorcing from my first marriage? Was a longer waiting period (iddah) required of me, and did my first marriage originally necessitate an iddah at all?

The preponderant opinion is that it is not permissible for a woman to marry before her invalid marriage is annulled or she is divorced. Marriage before the expiration of the waiting period (iddah) of divorce is void. You should have waited until your iddah expired (three menstrual cycles). Since you married before the iddah expired due to ignorance of the ruling, it is hoped that there is no sin upon you both as long as you were ignorant of the Sharia ruling. However, it is forbidden for you to continue, and you must cease.