For me, the only natural one is 'rent it out'.

Yes all these sentences are grammatically correct. The differences are subtle! Generally in British English the difference between renting and hiring is the duration of time and what is being rented or hired. So for instance in relation to property you would normally always use 'rent'. For equipment you would generally use 'hire' So you could hire a car for a week, or a bike for an afternoon. In the first sentence you can say 'lease it' or 'lease it out'. As the owner of the property you would generally 'lease it out', but the person who then lives in the property would 'lease it' from you. There is also I believe a difference between American English and British English as to how 'hire' and 'rent' and 'lease' are used, depending on whether you are the owner of the property, or are the user of the property for a duration of time

I think that while talking about real estate people use the verb "rent" "lease" is for cars or appliances and "hire" refers to contract somebody's services