If an airplane is rented to a customer on month to month contract basis rather than hourly rate with minimum per month, is this considered as a rental or lease? Is there a regulation that limits long-term rental? If a plane is rented to someone for a month, is there a regulation or policy in loan or insurance that tells the operator to maintain the airplane's operational control (knowing where the airplane is, etc.)? What's the line between lease & rent? We're talking about piston airplanes under 12,500lbs.
Since 14 CFR 91.23 covers only "large" aircraft (over 12,500 lb), your questions are beyond the scope of the Federal Air Regulations. You'd need to ask an aviation attorney familiar with the contract laws of the state under whose laws the contract is written to review that contract.