Lumber, drywall, shingles, flooring, cabinets, doors, siding, carpet, furniture, and ordinary cleanout junk are all fine in a Texarkana container. The limits come from Texas rules and from what the receiving landfill will take.
Texas handles asbestos, liquids, and other difficult material through the Texas Commission on Environmental Quality as special waste, which means it has to be authorized and routed to a facility that can accept it rather than slipped into a mixed load. Paint, solvents, fuel, motor oil, and pesticides all stay out.
The City's own refusal list is a useful guide to what needs a container and what needs a different plan. Its curbside program will not take appliances containing refrigerant, batteries, motor oil and filters, tires, vehicle parts, gasoline, paint, or anything over 200 pounds, and it specifically will not take concrete, brick, asphalt, roofing, or contractor construction debris.
Tires and appliances with cooling units are usually billed per piece rather than by weight, so declare them when you order. Mattresses are worth mentioning as well.
Cover the load, too. The City charges a fee at the gate for an uncovered load even when the same material would otherwise be free, which is why the code requires every construction container to be able to take a lid or tarp.