What is the ruling regarding the emission of semen during tawaf due to overcrowding and extreme proximity to a woman, and what should one who did this do to feel at ease?

What you mentioned about losing your mind is not apparent due to the precision of the description. You made a grave error by succumbing to lust during the circumambulation (Tawaf). You must repent and feel deep remorse, for remorse is repentance, and you must resolve never to return to such an act. If you had a choice in avoiding what happened, then repentance is obligatory. However, if you had no choice, such as due to overcrowding and being unable to control your lust, then there is no sin upon you. For details regarding the emission of semen while in a state of Ihram, please refer to Fatwa no. 36778. If it occurred after the second disengagement (from Ihram), then no ransom (fidyah) is required of you.