Translation:They are singing and washing a happy dog.
No. That would be, "he laujavat ja pesevät onnelliksen koiran". Partitive case objests are indicative of something in progress or continuing, so present progressive is the correct English translation. Simple present indicates a regular practice or something happening in the future, and takes an accusative case object. While simple and progressive present are equally correct in the abstract, the case of the object and any modifiers determine which is correct in context.