Heather gave an awesome answer and I just wanted to chime in. "설거지는 하고 자는 거지?" translates to "You're doing the dishes before you go to sleep, right?" in English. "to do the dishes" is "설거지 하다" and the reason why the verb ending -하다 changes to -하고 is because the speaker is asking if the listener will "do the dishes AND THEN go to sleep" so the idea is "and [then]". Hope that helps.

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'.