Which 3rd party libraries could be used with Kotlin Native? For example, if I want to write app that reads JSON from file and writes it to database, would I use Jackson and Hibernate or some C stuff?

Any library with C bindings could be used, see samples coming with Kotlin/Native, for your task you may want to use https://github.com/json-c/json-c and C connector for MySQl, i.e. https://www.mysql.com/products/connector/

I don't think hibernate will work as it uses cglib proxies - cglib/asm depend on the JVM