Ordinary household and construction debris is fine. The limits here come from New York State law and from what the receiving sites are allowed to take.
New York makes it illegal to put lead-acid batteries in the trash, with a civil penalty of up to $50 for each violation, and retailers have to take them back free. Electronic waste is under a statewide disposal ban as well, so televisions, monitors, computers, and printers have to be recycled separately.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos stay out of every standard container, as do propane tanks and pressurized cylinders.
The local rule to know is that the City's own waste site turns away building material from construction, demolition, alterations, or repairs, along with slag, rock, sand, brick, concrete, tires, large tree trunks, and carpet rolls over a foot thick. That doesn't mean you can't rent a container for that debris. It means the load goes somewhere else, which your account manager will set up.
Separate your yard waste too. The City won't collect it curbside if it has dirt, rocks, or construction material mixed in. Ask your account manager if any item on your list looks borderline.