Standard construction and cleanout debris is fine. The limits come from Florida law, from the City's own definitions, and from what the Lee County facility will take.
Florida bans certain batteries from the trash outright. The state Department of Environmental Protection says it is illegal to discard nickel-cadmium and small sealed lead-acid rechargeable batteries, or products containing them, and notes the prohibition reaches every resident and every business in the state.
Cape Coral's own rules keep hazardous material out of any container. The City defines hazardous waste to include radioactive substances, toxic chemicals, biological waste, flammable waste, explosives, acids, chlorine, oil, grease, paint, and paint thinners, and it excludes all of that from construction debris, from garbage, and from bulky waste.
Vehicles and liquids are out as well. The City's bulky waste category expressly excludes cars and boats, vehicle components other than wheels and tires, liquid waste, hazardous waste, and industrial waste.
Appliances need their own route. Anything containing regulated refrigerant is handled as a white good, collected separately rather than mixed into a container.
And as noted above, asphalt shingles now need a different destination than the county's main campus. If your job includes a shingle roof, say so when you book so the haul is set up correctly. Anything you are unsure about, ask your account manager before it goes in.