I'm fairly certain it would work over ethernet, but wireless would be cooler. Looks like Dish didn't load the wifi adapter drivers on the 4K Joeys.
If anybody wants me to try it over ethernet I will.
I would assume that works on the 4K joey as it does on the other joeys. But as stated before you either need a gigabit network or directly connected to the Hopper. With Cat5e or better.