Does the rest of the world know what's happening in Panem (do they even care)?

As far as the book and the movie are concerned, there is no rest of the world. It has also gone completely untouched by Suzanne Collins. The book sometimes implies that the rest of the world was destroyed, between wars and geological disasters (Panem is located in North America [Canada, Mexico, USA], but most of modern day coasts are shown as flooded, Sea level rise). But we only have Katniss's word for this. And Katniss is a slave in a despotic country, one that is reminiscent of modern day North Korea. Any education given to Katniss and other district citizens is highly controlled by the Capitol. Katniss doesn't even know much about other districts, let alone District 13 being real/surviving, because of Capitol restriction on knowledge. Katniss herself mentions this multiple times, that what she knows, is what the Capitol wanted her to know.

As the adage goes, history books are written by the victors. Whomever is in charge determine what's the correct information to give. See 1984, Interstellar, V for Vendetta, or any book regarding dystopian governments. Or you know, real life.

So we have an unreliable, undereducated narrator, and an Author that refuses to broach the subject. No answer can be given.

We can assume though, that outside nations do exist. After all, why spend any effort on maintaining and developing an active Nuclear arsenal when you have no enemies that you can't destroy with standard bombings? District 13 was, before the Dark Days and the Hunger Games, the designated military defense for Panem and the Capitol. After the districts rebelled 75+ years ago, and District 13's and the Capitol ended in a nonaggression treaty, allowing the Capitol to destroy 13 (they agreed to hide) and put down the rebellion, District 2 became the military hotbed ("The Nut", the bunker in the mountains that was buried in). The Capitol would not use nuclear weapons against the Districts because of how important the Districts are to it, resource wise. Nuclear attacks would be suicide. So the only rational reason would be to use as deterrence against outside military powers.