Does one have to keep separate ketchup bottles for meat and dairy? (and why or why not?)
It is commendable to use separate ketchup and mustard containers for meat and dairy but it's usually not required as long as it does not touch the foods directly. If you use one container you should use squeeze type bottles, so care can be taken that the bottle doesn't touch the food on the plate. Since it does not directly touch the food it does not become an issue unless there are small children in the home who will put the nozzle in the food leading to direct touching rendering the bottle Fleishig or Milchig depending on what it touched.
Source for it being an issue: Shulchan Aruch Yoreh Deah Siman 105:3 in the Rema, where he discussed pouring oil onto a candle made out of non-kosher fats. Both the Shach and the Taz discuss this idea of nitzuk chibur (pouring creates a connection).