Is the designated prayer area inside a mall considered a mosque, for which one is rewarded for walking to it and in which one prays Tahiyat al-Masjid, or is it a prayer room where praying is better than praying at home?

A mosque is any place that has been dedicated for prayer, whether small or large. It differs from a musalla (prayer room). If a place is prepared for prayer and made permissible for all Muslims, then it is considered a mosque because dedication is established by action just as it is established by word. So, whoever consecrates a plot of land and permits people to pray therein, it becomes a mosque. Therefore, it is legislated for you to perform Tahiyat al-Masjid (the two rak'ahs upon entering a mosque) when you enter it, and you will be rewarded for walking to it.