If the marriage has not been consummated, there will be no 'iddah for divorce and the wife will be entitled to half the mahr.
Consummating means sexual intercourse, and so there would be 'iddah if (1) there is intercourse, or (2) the husband's seminal fluid (somehow) entered the wife's vagina.
The first husband must redo the Islamic divorce, making sure all the requirements are met, and it would be best that he authorises a scholar to conduct the divorce on his behalf.
As for the woman, although her temporary marriage was not sinful, it could be that she is not permitted to marry or be with that second man she was with in the temporary marriage. This is something that she must refer to her Marja' taqleed about.
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