When mixing chemical solutions such as aqueous stains/solutons, what is the proper way of cleaning the containers used for the mixing. Is it appropriate to clean them like normal labware and rinse the contents down the sink?

Depends on the chemicals, many are not safe or allowed for discharge in sewer danges ( also th size and amount of materials)
Many need to be disposed of by putting in proper containers and having them disposed of by companies that do so according to law.