What is the ruling of Islamic law regarding a man who offered his wife to another man, and that person photographed her, then claimed to have touched her? What are the consequences of his actions? Should this man be reported, knowing that the wronged party has kept it concealed? And how does one repent to Allah from such a sin?

What you did is an ugly act and contrary to sound human nature, and you did well by repenting to Allah Almighty. If your wife is aware of what happened to her, she must also repent. As for this man who photographed your wife, he must repent and delete this picture. You acted well by asking him to delete it, and you have cleared yourself of responsibility in that regard. As long as he has sworn by Allah Almighty that he has deleted it, you have no choice but to believe him in that and entrust his matter to Allah Almighty. Covering a Muslim's sins is obligatory, and not informing that man of what happened and concealing it from him – if true – provides concealment for everyone.