Hey Liutke, don't panic (:
If you see an animation being influenced by other animations in an unwanted way, it means you need to put keyframes on the first frame on the bones that are being influenced until everything looks nice in preview mode.
To speed up things you can also just press ctrl+shift+L
to keyframe everything in one go. (the Cheat Sheet has other nice possible shortcuts if you're curious Spine: Cheat Sheet)
Regarding clipping in Unity, we need to ask Pharan who's the expert 😉