What do you mean by _running_ specific parts of the code? Isn't this a language problem (compile all dependencies (classes) to byte code, package .class files in jar files, define the main class, define the classpath, and then run)?
Some devs really like this. I never really encounter it as a problem because for any refactoring, I prefer to use IntelliJ's refactoring tools that ensure all code (that's in your project at least) gets rewritten in the same moment the refactoring occurs.
That's the same as Eclipse (e.g. rename method), right? What's the benefit of IntelliJ here?
(Edit: Some of the IntelliJ operations are distinctly superior to the Eclipse ones. But not all. In my experience I'd say they're about even here.)
For example, in Eclipse it's trivial to take an inner class and move it to its own file. In IntelliJ that doesn't seem to be an option.
And that's a breaking change if I do it manually, which is where the Problems panel is so useful.
Not forgetting TDD where in Eclipse I can write multiple tests for a feature before any code exists and run them before the unit I'm testing even fully compiles!
For example, in this screenshot one of the methods the test class invokes doesn't exist: https://paperstack.com/img/ides/eclipse_run_broken_code.png
2. For non-static class: Refactor > Move Inner Class to Upper Level
Then you can select class name and parameter name of formerly enclosing class.
Still, when a refactoring operation doesn't exist for a code change (even if it's due to operator ignorance) it's nice to have the Problems pane to tidy up with.