Crop dusting regulations in the US
To answer your questions:

A crop duster does not have to file a (normal) flight plan. They do have to file a special "operations" plan if they are going to be spraying chemicals. Notice of the intended operation must be given to the public by some effective means, such as daily newspapers, radio, television, or door-to-door notice. A plan for [the spraying] operation must be submitted to, and approved by appropriate personnel of the FAA Flight Standards District Office having jurisdiction over the area where the operation is to be conducted. The plan must include consideration of obstructions to flight; the emergency landing capabilities of the aircraft to be used; and any necessary coordination with air traffic control. 

All aircraft must maintain a height of at least 500 feet above ground or objects on the ground, like a house. (Minimum Safe Altitude) A crop duster can go below this height only to operate to apply chemicals and for no other reason. The exact text of the exception reads as follows: "No person may operate an aircraft over a congested area below the altitudes prescribed in part 91 of this chapter except during the actual dispensing operation, including the approaches and departures necessary for that operation."

You cannot prevent someone from flying over your house.

If someone is flying over your house at less than 500 feet, you can file a complaint with the FAA by calling (703) 230-7664. Please observe the following:

• Identification - Can you identify the aircraft? Did you record the registration number which appears on the fuselage or tail? (On U.S. registered aircraft, that number will be preceded with a capital "N".)

• Time and Place - Exactly when did the incident(s) occur? Where did this happen? What direction was the aircraft flying?

• Altitude - How high (low) was the aircraft flying? On what do you base your estimate? Was the aircraft level with or below the elevation of a prominent object such as a tower or building? Did you obtain photographs? Are there any witnesses who could confirm your estimate - do you have their names, addresses, telephone numbers?

• Supporting Evidence

• Witnesses, Police

• Photographs