It is not within the normal daily occurrences for a lender to send an inspector to check the property. For your own protection, it is always best to hire your own inspector. Most RE professionals have 1 or 2 trusted inspectors that can be recommended to you.
Everything in real estate is negotiable.
In the Tampa Bay area market, 8-10% is pretty standard but the final negotiated amount depends on the level of service that the property manager is going to deliver to the homeowner. I know that that percentage may seem high but, as real estate professionals, it is our job to make sure that many things come together at once. It is also our responsibility to find a solid tenant, collect rents in a timely manner, maintain the property to the owners' standards while following state and local laws, begin eviction precedings if needed, etc. The landlord/tenant laws in many states leave little to no room for mistakes. If an owner "messes up", the penalties can be quite high monetarily. RE professionals understand all the ins and outs of these laws and make things go smoothly.