Yes, '하고' means something else besides 'and' or 'with' here. '하고' is from a verb '하다' which means 'do'. To be specific, '하고' is a short form of '하고나서' which means 'after doing something'. That's why '하고' here can be confused with '하고' meaning 'and' or 'with'. Therefore, '설거지는 하고' is actually '설거지는 하고나서' and it is translated into 'after doing dish washing'. In other words, it is 'after washing the dishes'.