Should someone entering the mosque pray "Tahiyat Al-Masjid" (Two Rak'ahs for greeting the mosque) while the Adhan is being called, or should they wait until the Muezzin has finished?

It is best for the listener to stand until the Adhan is completed, because of the Prophet, peace and blessings be upon him, saying: "When you hear the Mu'adhin, say as he says." The Prophet would respond to the Mu'adhin word for word, except during the Hay'alah (come to prayer/come to success), when he would say: "There is no might nor power except with Allah." It is narrated in a Hadith from Omar that whoever responds to the Mu'adhin from their heart will enter Paradise, and this is a great virtue. Once the response is finished, one should pray the greeting of the mosque, to combine the two Sunnahs of responding to the Adhan and greeting the mosque.