As long as the Pokemon isn't in a gym match, you can use it in more than one battle. As for the ref leaving, I just came back from a disappearing act, so I wouldn't know if you had a ref or not. Sorry if I sound kinda mean. :x
There is a default DQ time, but it's not always enforced, and depending on the ref, they may choose to run with the default DQ time or not implement it at all, unless someone requests it.