I have categories in two languages. You explained how I could select only categories in one language in the menu-settings.
I also want only the categories from one language in the Search Module. Is this possible?
You can use one instance of the module for each language site.
Then in the module's parameters, if you disable the category search on the module, then assign the module's ItemID parameter to the correct language specific menu item it will only search the selected category for that menu item.