Should the zakat on the wealth of one who owns residential buildings for rent be on the collected rent, or on the capital and profit? And is zakat obligatory on vacant land not prepared for sale, or on a private residential home?

Apartments prepared for rent are not subject to Zakat (definition). The same applies to empty land and a house prepared for dwelling if they are not intended for trade. This is agreed upon by the majority of jurists and Fiqh councils. However, if the rent is collected and added to one's wealth, and it reaches the nisab (definition) and a full year passes on it, then Zakat becomes due on it at that time.