Are the first and second 'Umrahs valid, despite doubt concerning the emission of something from the front private part during ihram and tawaf, and is it obligatory to ascertain the emission of mathy (pre-seminal fluid) upon feeling aroused?

The two 'Umrahs mentioned are valid, Allah willing. The sensation of something exiting the front private part has no effect unless it is a whisper from Satan (waswasah) or a doubt, and you have repeated the ablution (wudu'), which is a condition for the validity of the circumambulation (tawaf). Doubt concerning the garment being touched by impurity has no effect, for the original state is purity, and "certainty is not removed by doubt." What you felt in the second 'Umrah regarding the arousal of desire also has no effect, for the Prophet, peace and blessings be upon him, said: "One should not leave until he hears a sound or finds a smell." It is not necessary for one who doubts the exit of madhy (pre-seminal fluid) to look at his garment; rather, he should ignore the doubt. And the sheep slaughtered in Makkah is a charity for which its owner will be rewarded.