Build Minecraft mirroring?

How do I get two doors connected to Redstone? So when I open one door, the other one opens automatically.

You could place redstone over the door and then activate it with a button

Lay a redstone line one block under the doors and then connect it with a pressure plate / switch / button etc.

Is there a way to build it underground and without a button? So with observers?

Yes you make an observer under one door with the story looking at the door then the other with the side where redstone is issued to the other door and then put redstone under it