Hello @Irish3538
Yes, you can connect multiple secondary/point units to a switch that's connected to the primary/router unit's LAN port. I would recommend sticking with inexpensive, unmanaged Ethernet switches, though – some smart/managed switches can be made to work, but they may require configuration first related to loop detection.
As for your bonus question, it does look like the TP Link RE550 can be set up in an access point mode fed from an Ethernet connection. Just from a connectivity standpoint, while you can connect an Ethernet device to the second port on a wired secondary Nest WiFi Pro unit, I personally prefer using a local 5-port switch at the end of the run that can then feed both the Nest WiFi Pro unit as well as other Ethernet gear in the area.
But, beyond the Ethernet consideration, I would note that adding another brand of access point to your system should be done carefully. If it's configured with a separate network name+password, it won't cause noticeable issues. But if it's configured with the same network name (SSID) and password as your Nest WiFi Pro network, some client devices may have trouble handing off between that access point and the Nest WiFi Pro units. Either way, since that TP Link unit will be a bridge, any clients connected to it will all be part of the same network segment as all of the WiFi clients connected to Nest WiFi Pro units and anything connected to the wired Ethernet network through that switch (and any other leaf switches connected to it).
