You can easily check if you have Space Age :space-age: enabled by looking at the load or save game GUI.

Just look at your savegame and check out the mods list. If it contains "space-age [version]" it means the Space Age DLC is enabled on your save.

In order to visit planets other than Nauvis you ...

I have researched everything I have Space Age. But I can not see or explore Space Age! Nauvis is ready.

Do I have Space Age? Where can I see that? '
Can a peaceful mode Space Age prevent?