Not sure why they would want a fasting and a non-fasting on the same day. The second one is going to be potentially influenced by whatever you eat for lunch. Although I guess they could get some information.
Don't worry too much about the ketone test, it's just if you are a Type 1 in theory you should have raised ketones even if you do have some limited insulin production left. If you are not currently getting any diabetes symptoms it's unlikely you are in
DKA. (Northener/Alan will correct me I think)