>>2043254 (OP)
Connecting two computers directly to one another only possibly establishes a link-local connection, if both are setup to automatically assign themselves an IP address within the link-local designated range; if they have done this, and it still doesn't work, then I'd suspect whichever games you are trying to play just haven't been programmed to work over link-local.
https://en.wikipedia.org/wiki/Link-local_address
>>2043447
>>2043933
Using only a switch with computers attached to it won't work, as you need a router, or more specifically a designated computer that can forward packets, not just receive them, and every computer needs that router to be set as its main route.
Here's the simplest setup that I can think of for hooking a bunch of computers up together in a LAN to play together:
1. A router enabled with DHCP
2. Optionally, an unmanaged switch to connect to the router if the computer acting as the router itself lacks the necessary amount of ports to connect all of the computers to
3. Functioning DHCP clients setup on all of the computers
4. Make sure all of the computers are connected to the LAN-facing ports of the router and/or the switch
After that's all set, you need to make sure that the computers connected to this LAN have private IP addresses that are within the same subnet as the router. Firewall issues probably only reside with the client computers, if any at all; port-forwarding is unnecessary in a LAN setup like this given how most routers are configured by default
If possible, use tools like ping and netcat to help diagnose issues
https://en.wikipedia.org/wiki/Private_network#Private_IPv4_addresses
https://en.wikipedia.org/wiki/Ping_(networking_utility)
https://en.wikipedia.org/wiki/Netcat
https://simple.wikipedia.org/wiki/Computer_network
https://simple.wikipedia.org/wiki/Internet_Protocol