Donnelly's order is a temporary emergency stay until the Supreme Court can decide. Trump's administration will not go to another district court, because, as you point out, that would cause unnecessary confusion. Rather, they will go to a higher court. Note that, as DHS points out, this does not allow people stuck in airports to enter the United States: they just can't be deported back to their home country. Additionally, it does not affect people who haven't flown here yet: they are still banned. It just means the people stuck in airports are in limbo: they can't be forced back to their country, but they can't enter the United States (ie, can't get past Customs and Immigration to leave the airport) either.

Judge Donnelly can issue a stay (i.e. [a temporary stop to a legal proceeding](http://legal-dictionary.thefreedictionary.com/stay)) because she is a federal judge, and thus

> have "federal question" jurisdiction, which means that federal courts
> will hear cases that involve issues touching on the Constitution or
> other federal laws. [Source](http://www.catea.gatech.edu/grade/legal/juris.html).

Donnelly's decision was based on "irreparable harm" (I might have incorrectly said "immediate harm" earlier). Suppose Donnelly allowed the deportations, the refugees were killed after arriving back in their country, and a higher court then decided the deportation was illegal. At that point, there is nothing the higher court can do to repair the damage caused to those deported.

Conversely, allowing them to remain (detained inside an airport, not free to roam the country) has very little risk. If a higher court rules they should be deported, they can be at that time-- the delay in deportation causes no irreparable harm.

You could argue the detention itself is a form of harm, but courts have generally held that you can compensate people for illegal detention with money. In other words, detention is harm, but not irreparable harm, at least in this case.