GS305E and linking two home networks/IP ranges

In general, only one device - with it's WAN/Internet port - can be connected to a ONT.

The GS305E is a simple unmanaged switch (yes!) with some very basic config options for defining VLANs.

The crystal ball is not in the office on a weekend, so hard to guess whatever "bizarre" IP addresses you are talking of. The IP address you experience or see however could explain what is (not) going on there.

In blunt theory, you can of course operate multiple networks on different IP subnetworks when properly configured on a switch for dedicated VLANs. Said this: You can't connect multiple LANs and IP subnets without properly isolating on VLANs, and access ports (associating unmanaged incoming frames with  singe VLAN only) can only configured to work on a single VLAN.

No matter if you think about Plus switches, Smart Managed switches, or Managed switches: DMZ is a term known on NAT routers, not on switches.

Talking of Google WiFi or similar the Google Nest WiFi Router, there is always only one system connected what makes up your Internet connection. More coverage can be added by Google Nest WiFi Pods, purely making a wireless backhaul. The Google Nest Pro devices are adding dedicated LAN port(s) for locally connecting devices requiring fast and reliable network connection, e.g. for adding say a NAS, a gaming station, a PC workstation local to each Google Nest Pro device.

Where and how are you connecting this poor little GS305E so it's apparently causing a connectivity problem on your network please? Yes, the GS305E does have a feature to avoid and blocking networking loops for the case the kids create a network loop. It does not crash - it would just shut one of the ports involved in the loop.

Said this: No, you can't direct connect to networks together and magically get two or more dedicated networks.

If looking for support, please always create a list of the exact devices involved, the connections (ideally a simple drawing).

Last: Keep in mind most of these Plus switches don't have a management VLAN. The uC with the simple IP stack can and will talk to any data flowing untagged through the switch. So it's not possible to properly isolate the config IP from multiple VLANs, even if configuring dedicated access ports for different VLANs.