Lumber, drywall, roofing, flooring, cabinets, trim, doors, carpet, furniture, and ordinary cleanout junk are all fine.
King County publishes a long list of disposal restrictions for its own facilities, and that list, not the hauler, decides what your load can contain. Major appliances are the one that surprises people most: refrigerators, freezers, stoves, washers, dryers, water heaters, furnaces, and air conditioners are not accepted for disposal at all and have to be recycled separately. Dishwashers made mostly of plastic are the only exception.
Asbestos needs county approval before anyone will take it, so it belongs with a licensed abatement contractor. Big dense objects over two feet in every direction and heavier than 200 pounds, such as concrete blocks, safes, and stumps, also need clearance first. Vehicle and other rechargeable batteries are refused outright, and liquids have to be sealed and kept under five gallons per visit.
Tires are a hard no from businesses, and residential customers are limited to four passenger car tires. Large truck tires are not accepted anywhere in the system.
Hazardous waste, paint, solvents, and fuel stay out of every container. The county takes household hazardous waste free from King County residents at its Auburn-area station, which closes that service on Wednesdays.
If anything on your list sounds like it might be on that list, raise it with your account manager before loading day.