Couple of things. It's late so I apologize if I missed it in the replies, but are you a new Dish customer or did you have a Hopper or other Dish receiver before the H3?
If new, maybe the installer used the wrong hub with the setup? Wrong type of cable? Wired wrong? Connections not tight? Something in the install to where the signal is being affected. Especially if they have not done an H3 install before. Since you said the Joey is poor too, then it kind of leans toward install rather than equipment most likely.
Out of the box the picture should be clear HD. Have the H3 and one Joey connected to Sony HD TVs (not 4K) and one Joey connected to a Sony projector. Solid pictures. I've been a long time Dish customer so I had the proper cabling in already and the installer knew to replace the old hub with the new one.
This was posted for the wiring and equipment (you're probably going to be the first setup on the top of the second page if only one Joey):
http://www.satelliteguys.us/xen/threads/hybrid-lnb-wiring-diagrams.357083/#post-3815139
And this for the cable type:
http://www.satelliteguys.us/xen/threads/3-ghz-or-not.357315/
Trying to offer some solutions since you said you had the TV output settings right and know the HDMI cables/TVs are good quality from the other TV services you mentioned.