To what extent is it valid to stipulate that a Muslim lower one leg from the other, or stand up in reverence to the Adhan or when reciting the Qur'an, and is this considered among Islamic manners or newly introduced customs?

It is permissible to place one leg over the other when hearing the adhan (call to prayer) or reciting the Quran. The Prophet, peace and blessings be upon him, recited the Quran while his head was in the lap of Aisha when she was menstruating. As for standing out of reverence for the adhan, some jurists have deemed it recommended, and a hadith to that effect has been reported, though Al-Albani declared it weak. The established Sunnah indicates that it is the mu'adhin (caller to prayer) who stands for the adhan, not the listener.