TiVos can handle switches just fine. The TiVo rep should have said that TiVos on different LANs cannot communicate (although that may be possible with VPN, but that is a separate issue...)
You need to have the Roamio and Premiere "phone home" so that the Mini can see them. The fact that the Mini shows the red circles means that it can actually see the other TiVos, but they don't know the Mini is there. You may also need to go into your TiVo account online and be sure the right checkboxes are checked for sharing between all of your boxes.
MoCA won't work if the Mini is the only MoCA device on the coax -- you need to have either a Roamio Plus/Pro or a Premiere XL4 or an external MoCA adapter to create the MoCA network (or if you have FiOS, their system will create the MoCA network for you). MoCA is simply technology to bridge ethernet signal onto the coax along with your video signals. It tends to be used by people who can't run Cat5e/Cat6 cables to all of their devices.
I'd suggest going back to all ethernet for now, and have all boxes "phone home" to see if that gets rid of the red circles. Assuming everything has been able to reach the internet and get the latest updates and guide info, they will be able to stream over ethernet once they all can see each other.
If that doesn't work, you can try MoCA, but that may require additional equipment and expense that probably isn't necessary. One thing that could cause problems is using WiFi -- if your network is all hardwired, then it should work very well once everything is configured correctly.
Oh, just saw that you mentioned MoCA setting on the Roamio -- if it is a Plus/Pro, then you can set it to "Use this DVR to create a MoCA network" and connect the Mini to coax. One MoCA device needs to have an ethernet connection as well as a coax connection so that it can bridge from the LAN to the coax.