Connecting via a CAT cable should be fine, but remember the entire system will be limited by the specs of the weakest component. So try to be sure the device putting out the signal in the barn is a XE75 or better in specs. 

  @Christianbrite yes, assuming you have some power in your barn, you can add any model of Deco into your mesh.  It can connect by wifi or by wire to any of the other Deco.

If there is no power in the barn, you need a POE model of Deco, capable of being powered via the Ethernet cable, and you will need a POE injector in your home.

Some "advanced" features of the Deco mesh are supported only by some models (plus hardware and firmware levels) of the Deco units.  Such advanced features might not be available in a mixed environment (mixing models, hardware/firmware levels).

  @WYLDSTALLYNS 

Thank you, I'll be purchasing an exact match to the 3 nodes i already have.  

  @yves_b 

Thank you, I do have power in the shop/barn.