I listed two clear counter examples to claim #2, C++ and Javascript. I'm even tempted to put C in that list, because C was already popular before Linux.
What was Python's killer app?
What is Rust's killer app?
Go's?
Kotlin's?
Java is a counter example to your claim #3 since it not only does it run on all OS'es but you can also develop it on all OS'es.
Killer app is the wrong phrase. Rather every new language needs a competitive upside over existing languages to be successful. At which point your proposed questions have easy answers.
What was Python's killer app?
What is Rust's killer app?
Go's?
Kotlin's?
Java is a counter example to your claim #3 since it not only does it run on all OS'es but you can also develop it on all OS'es.