What is the ruling on promising upon the Quran to stand for prayer immediately upon hearing the Adhan, and how should one act when the Adhan coincides with eating, sleeping, or being in the street?

If what you uttered was an oath on the Quran, then you must fulfill it, and expiation is incumbent upon you if you break it. However, if it was merely a promise, then fulfilling it is recommended but not obligatory. If it implied a commitment, like a vow, then it is obligatory to fulfill it. Performing prayer at the beginning of its time is among the best deeds, and there is no harm in delaying it as long as its time has not expired.