While the server side is still growing and obviously not as huge as Java etc., Dart definitely is being used on the backend. For instance, https://pub.dev/ itself is written in Dart: https://github.com/dart-lang/pub-dev . :)

The language, like any other, does have some weaker parts, but in general is _very_ nice to write in - thanks to both the language itself and the toolchain built around it.

The apps compile to native binaries for production and are run in Dart VM during development. The SDK is multi purpose. You can write "scripts", you can write CLI apps, servers and with Flutter - the rest of them.