Personally, I don't even ask many questions of your last landlord. The reason? if you were a bad tenant, then that landlord is likely to give you glowing reviews just to get rid of you!!!
In most cases, the new landlord would be expected to ask questions like: did you pay your rent on time, were you ever late? how did you maintain the home? were there ever any issues during your tenancy? They might also just confirm whatever you had listed on your rental application as to the duration of your tenancy and things like that.
I can't imagine the new landlord asking personal questions, and can't imagine what they might care to ask personally except those related to your new rental application.
A landlord can select a tenant based on two criteria: your ability to pay, and your ability to maintain the property in the condition that the landlord expects. That assumes that you did not violate the lease by such things as having loud parties, extra people living there that weren't on the lease or things like that....but again, I never expect the last landlord to be my best reference for you. They have too much at stake on whether you stay at their place or move to mine. Hope that helps.