I can handle some early adopter pain, but I want to focus mostly on writing an application.
We work on new API reference for both backends and big part of current stdlib will be available in the future for JS backend too.
2. I don't know about any project where Kotlin used in both side.
In IDEA you can create two unrelated projects and set up a module in each of these using the same source directory, but different targets.
Additionally You can try do it by ant or maven build by sharing source dirs between different ant/maven projects.