Am I still in my first waiting period (Iddah), and was what happened between my husband and me permissible (halal) or forbidden (haram), or do we start the waiting period again after the second divorce? And are my ex-husband's visits to me after the divorce permissible (halal) or forbidden (haram)?

If he divorced you with the first pronouncement without compensation, then it is a revocable divorce (raj'i), and his taking you back during the waiting period ('iddah) is valid. The second divorce is effective, and he also has the right to take you back during the waiting period. The statement of the marriage officiant (ma'dhun) that it is an irrevocable divorce (ba'in) is not to be considered; because the default is that divorce is revocable unless it is the third pronouncement, or divorce before consummation, or divorce for compensation.

However, if the divorce was for compensation, then it is an irrevocable divorce (ba'in) in which the husband does not have the right to take you back. The residence of a woman irrevocably divorced with her ex-husband is not permissible, except out of necessity with the assurance of no temptation (fitnah) and her independent dwelling in a part of the residence. In this case, his taking you back without a new contract is not valid, and you must separate from him and resume the waiting period ('iddah) due to the intercourse.