1. Can I do second morning hand washing in the kitchen sink? 2. Can I use any towel to hold the cup, for example, kitchen towel that we normally use to dry hands with? Thanks.
Yes the second washing is not important by your bed or right when you get up. As well the washing should not be done in a bathroom. After you are all dressed and ready to go you go to the kitchen and take the cup with towel and wash a second time with a bracha. The towel can be any towel that is present and clean for wiping hands even if used prior for regular drying.