If I have an android and jvm("desktop") target, can I make another target that both of those will use for common jvm code?

"Sharing code between JVM and Android" is "postponed for later" on the kotlin roadmap

You can do this, but the IDE support probably won't work well. It'll build fine from Gradle if you can deal with red code in the IDE.

that can be quite challenging when code completion and navigation doesn't work for any JVM APIs

Agreed. There's various workarounds but it doesn't really scale all that well. But it's not impossible to do.