Same as the US.

Wait in a central waiting hall until the train arrives, at which time a big screen shows what platform the train is on. Then you and everyone else rush to be first to get to the single escalator going down to the platform. A conductor checks your ticket at the top of the escalator. That's how it works in NY. It's also approximately how it works in China, unfortunately (though their stations put Penn Station to shame in almost every other way).

That’s because there isn’t room on the platforms under Penn Station. And it’s very common in Europe not to announce the platform until 10 minutes or so before. What Amtrak could do though is have actual reserved seats on their all reserved trains.

It depends on where you are in Europe. In Germany and Switzerland, the platform is announced up to a year in advance, and you can go and wait there for your train. Sometimes you can even buy a coffee directly on the platform while you're waiting.

Furthermore, most tickets have assigned seating so there's no race. Also tickets are inspected when the train is moving, not on the platforms.