Hi,
How exactly are you using the 360 environment? How is your scene setup? Please attach some screenshots or a simple scene to understand it better.
By default, the environment is set to be a 10000cm radius sphere around the scene origin (Developmental and Experimental rollout in Render Settings > Performance), and the depth of field works for this - the environment is blurred just like an object 10000cm far from origin.
You can use the environment with "Dome mapping" and set a custom radius for it, to change the distance from camera and effectively blurring.
Shadowcatcher work just fine too, I have tested it.
As a note, lowering the F-Stop value will increase the depth of field effect.
If you want to use Depth of with a Spherical 360 projection camera, that is not supported.