Was the Prophet (peace be upon him) teaching the Companions how to pause on words like "نعمة" (grace) with a silent "hāʾ" or a silent "tāʾ" according to the orthography, and did he delve with them into such minute tajweed details?

The pause and resumption (waqf and ibtida’) in the Quran is a sublime art through which the performance of recitation is known. The Companions, may Allah be pleased with them, used to learn it just as they learned the Quran. This is indicated by the hadith of Ibn Umar, who condemned those who recited the Quran without knowing what it commands and forbids, and where one should pause. Ali ibn Abi Talib, may Allah be pleased with him, clarified that tartil (recitation with measured pace) is the beautification of letters and the knowledge of pauses. The learning and diligent care for it has been continuously transmitted from the pious predecessors.

As for how the Prophet, peace be upon him, taught these subtleties, the Arabs would naturally pronounce speech with tajwid (beautification) due to their innate eloquence and fluency. When foreign influence entered the tongues, the rules of tajwid were established as a description of how the Arabs used to pronounce words. Tajwid is considered an intrinsic characteristic of the Arabic language, and whoever abandons it falls into obvious error (lahn jali).