Are the three divorces counted when the word "divorced" is repeated three times in a moment of anger? And is a verbal reconciliation during the waiting period sufficient to take the wife back, or does it require a new contract and dowry after the waiting period ends, especially since the divorce has not yet been officially documented?

If by repeating the divorce you intended to affirm the first pronouncement, then it is counted as a single divorce. You may take back your wife before the expiration of her iddah (waiting period) without a new contract or witnesses. The raj’ah (return) occurs merely by uttering the words with the intention, and your wife returns to your marital bond. However, if the iddah has expired, a new contract is necessary. What is considered is the husband’s utterance of the divorce.