What is the ruling on a slight exposure of the 'awrah during prayer, such as the appearance of a part of the foot, and does this include the sole of the foot? And will Allah hold one accountable for the minutest details in the event of not having suitable socks available?

Scholars differed regarding the obligation for a woman to cover her feet during prayer. The majority obligated it, while Abu Hanifa, Ibn Taymiyyah, and Ibn Uthaymeen held the view that it is not obligatory. You may follow the opinion of the Hanafis if you suffer from intrusive thoughts (waswasah) or find hardship in adhering to the majority opinion. According to the majority opinion, the uncovering of a small portion of the ‘awrah does not invalidate the prayer as long as it was not done intentionally. It is recommended to cover them with customary clothing or socks. The uncovering of a part of them after the prayer does not cause harm, and there is no need to wear winter socks in summer. Your claim that summer socks are unsuitable stems from waswasah.