Answer ... Yes, an appeal can be taken from a ruling recognising or enforcing a foreign judgment, or a ruling denying recognition and enforcement. For example, in DeJoria v Maghreb Petroleum Exploration, 804 F 3d 373 (5th Cir 2015), the appellate court heard an appeal from an order denying recognition under the Texas Recognition Act.
Answer ... The party taking the appeal can seek a stay pending appeal to preserve the status quo. A stay turns on factors that include irreparable injury, likelihood of success on the merits and where the public interest lies.