My Linksys router should arrive any time now, and I already downloaded the mesh firmware.

Should I connect one of its LAN ports to my existing router, or do I connect to another NIC on the PC so that the mesh is on its own IP network? If the mesh should have its own network, should I enable IP forwarding between the NICs or connect the Linksys's WAN port to my existing network (or neither)?

Short answer: Only WAN port is designed to connect to your home network.

Long answer: Only the WAN port is designed to be the interface to other non-mesh networks. Generally home networks are 192.168.x.x addresses or 72.16.x.x addresses (both class B). The mesh is a 10.x.x.x network (class A). In setup on the mesh node-gateway, check the gateway box. This gives all computers on the mesh access to your home network and internet. This also means you have responsibility for yourself and others to not use "https" (encryption) and other activity that violates our fcc licensing on RF.

If you don't check the gateway box in setup, then only you on that mesh node has access to the internet and there's no related RF traffic to be concerned about.

I can see a problem here. I already use 10.* and break it up into several subnets. It would be a royal pain to migrate to Class B or C VPN IPs (even though 192.168.* does offer more than enough space for me to do so).

My Pentium 3 almost never needs network access anymore. I could connect the Linksys to it. The Pentium 3 box would have the mesh, but be isolated from the LAN and the Internet.