Is it permissible for the pilgrim to perform Hajj before fasting the fidyah for transgressions committed during his 'umrahs, and does he remain in a state of ihram until he finishes the fasting?

The prohibitions mentioned in the question—such as shaving hair, clipping nails, using perfume, and wearing sewn clothes for men—are part of the expiation for harm (fidyat al-adha) where one has the choice (fasting, charity, or sacrifice). If someone commits them knowingly and intentionally due to a need, there is no sin upon him, but expiation is required. If he commits them without need, he must repent and offer expiation. However, if he commits them out of ignorance, forgetfulness, or compulsion, there is no sin upon him and no expiation. These prohibitions do not affect the validity of the 'Umrah, and subsequent 'Umrahs are valid. He may fast as expiation at any time and in any place. He may perform Hajj before offering the expiation, but it is better to pay it immediately to clear one's conscience.