Yes, it can. But if the current owner secures a tenant and decides to sell the house in the future while the tenant is renting it, the new owner will be legally required to assume the existing lease at closing. In addition, the new owner will then have to honor the existing lease until the expiration date.
In some states, executors may rent out a property under the state's probate laws. In other states, an executor must seek permission from the court. However, there is nothing in the law that specifically prohibits renting out property while it works its way through the probate process.
Yes, of course. Probate can last a while so it would not be a bad idea to make some money before it sells. Just make sure the details of the rental are known to the probate lawyer when it comes down to sell the property.