Did weapons exporting actively happen before the modern era? Were there arms industries that produced weapons and were they usually owned by the state?

The short answer was yes, there were definitely arms and armor exports in the premodern world. How the industries were organized depends on period and region.

Here are some previous answers about late Medieval Europe:

I covered equipping late medieval and early modern armies here
Dr. Tobias Capwell of the Wallace Collection talked about this in an AMA