Managing someone else's rental property for compensation triggers a real estate or property management license in most states, and the stay length usually has nothing to do with it. What matters is that you're renting, collecting rent, or negotiating leases on behalf of an owner for a fee. Some states have narrow exemptions for on-site salaried managers or for owners managing their own property. Whether your specific arrangement needs a license is a question for your state's real estate commission or a licensed attorney there, and I'd get that answered in writing before you take a placement fee.
The month-long exemption you heard about is a different rule. Stays of thirty days or more often fall outside the short-term rental and transient lodging ordinances that regulate nightly stays, and outside transient occupancy taxes in many places. That's about how the property is regulated, not about whether you personally need a license to manage it. The definitions of transient vary by city and state, so check the actual ordinance rather than the general rule.
On insurance, a standard landlord policy is written for an annual unfurnished tenancy. Three things typically need attention on a furnished corporate unit. The owner's contents, because 10,000 or more of furniture per unit isn't covered by a policy that assumes the tenant owns the furniture. The occupancy pattern, because rotating tenants on shorter leases can be treated differently by the carrier. And liability, because more arrivals and departures means more slips and more claims exposure.
On your side, you'd carry general liability and, if you're handling client money, whatever your state requires for trust accounts. Errors and omissions is standard for anyone managing property for others. The corporate clients themselves will often ask for a certificate of insurance naming them before they'll place an employee, which is a practical reason to have this sorted before you go looking for business rather than after.
