The other thing that is odd is that they mentioned their mother's illness took a turn for the worse and they were communicating with only the CEO via phone. It seems unlikely that you only let the CEO know at the beginning of your absence and then don't let him/her know when you will actually be coming back. It also seems unlikely that the company would accept your resignation, post the position, interview, make an offer, and have the new person start all within 6 weeks. Perhaps it is just me, but that seems like an aggressive timeframe to get all of that done.