My I strongly suggest you ask for a CSR in the USA to get you set up, especially with an install as COMPLEX as yours. The offshore CSR's just aren't very good at complex installs. Asking for a USA CSR I think it will greatly reduce the chance of misunderstanding and disappointment that can occur if using Chat or overseas CSR's. As stated above, if you think you would want TWO Hopper 3's, then that won't be possible until January at the earliest.
One other point, the key is not only that the H3 has 16 tuners, but how many streams from the Hopper3 at one time would you need (meaning how many people may watch LIVE TV or DVR recordings all at the same time. There is a limit to what the tech can provide in that sense. I forget the max number of streams one H3 can handle, but someone on here will know and chime in. This relates to if ONE H3 is sufficient or if you might like TWO H3's.
As to using your LAN ("Ethernet" Cat cable) to stream to your Joey instead of the Hopper's MOCA network: that is NOT an officially supported method, and Dish can shut that down at any time. However, if you want to take that route for one of the Joeys and understand the risk, then go ahead. However, the wireless Joey is pretty robust and uses its own sub-network; that would be the preferable method
FWIW, since you have cat5 cable (access to your LAN) at you further location, you could use Dish Anywhere to access LIVETV (ALL channels) and DVR, but you would have to interface from either a mobile device or laptop/PC, but it will give you a separate full experience with a bit of delay on commands.
FWIW#2, (what I do on occasion) You could get a Slingbox and "Sling" your content using your LAN. The SlingPlayer app is free and while the app is available on Roku (at the mercy of your upstream and downstream), the best experience in HD is to use the Amazon Fire at the playback TV for full high quality HD and using your LAN exclusively fro best PQ.
The two FWIW above are solutions I use for access at odd places in the house were I do not want (or need) another Joey and the extra $7 per month, but good enough for my occasional use. Of course, both of those FWIW work OUTSIDE the home at remote locations for viewing anywhere you have access to the internet.
Good luck, and let us know what you finally get.