Laws regarding billboards and advertising are very local in nature and are typically handled under city/county zoning ordinances. Start with calling your local county zoning office. They will tell you the city/county laws regarding your particular residential zoning overlay, if city or state laws supersede county laws, and recent changes in law that might matter and if the sign might be grandfathered.
There can be different types of "residential" zoning and the city/county will tell you this; some allow limited commercial use and signage, and some don't. The housing subdivision you are in may also have covenants; you'll know if there are covenants if you received information when you bought property in that subdivision.
1,2,3,4,8: These depend on local laws.
5: Very generally speaking, land owners typically do not have absolute rights to land usage; that is the rationale behind zoning laws (among others, like health and public safety, building codes, national defense, etc.), because some types of land usage impact adjacent users and the general public.
6, 7: Potential consequences include fines and requirements to take the billboard down, but again, those possibilities are very localized. The size of the billboard could come into play; again, this will be very localized. Some signage may be grandfathered, too.
In order for the city/county to look at the situation and possibly take action, you may have to file a written protest with the zoning office; they would help with the process. You may have to present your case at a public city council or county commission meeting, but that basically involves saying such and such is happening and you want the city/county attorney to look into relevant laws. It would help your case if you had a list of names of others in the area who are also unhappy about the billboard. I doubt you will need legal representation to lodge a protest, but if it comes to that, Google for free legal aid in your area.
If the city/county attorney won't take action (which is possible, as this involves prosecutorial discretion as to if the city/county wants to press the issue with the landowner), you can look for free legal aid in your area and consider your options.