Which mock battle?
Do you mean the mock battles you start with a red icon on the screen of the command center building? The battle requires that you constructed the arena on the country overview map. The icon for it is located near Calbridge Stronghold.
The archery also can be built on the country overview map. It's slightly east of Aldlyn.
The quest about different climate armors is given to you by the weaver Sehtia in the work area south-west of Central. It also has to be built on the country overview map first, and then you can enter it on the normal worldmap like other locations.