Routers only route between LANs. You can have a fully functional LAN, including DHCP, without a router. You only need a router if you need to be able to communicate with other LANs.

When you refer to "Link Local," I think you are really meaning APIPA. This will actually determine if a host is trying to use an already used link-local address and change it.

Frames do not get "directed towards a specific IP address;" frames have MAC addresses which a switch will send to a specific switch port.

Whether or not you have DHCP on a LAN, the LAN can function normally, and printers, etc. can be discovered the same way they would if DHCP was on the LAN, assuming the devices were assigned IP addresses in the LAN network.

Routers only route between LANs. You can have a fully functional LAN, including DHCP, without a router. You only need a router if you need to be able to communicate with other LANs.

When you refer to "Link Local," I think you are really meaning APIPA. This will actually determine if a host is trying to use an already used link-local address and change it.

Frames do not get "directed towards a specific IP address;" frames have MAC addresses which a switch will use to send a frame send to a specific switch port. You are confusing layer-2 frames and layer-3 packets. A LAN is a layer-2 concept.

Whether or not you have DHCP on a LAN, the LAN can function normally, and printers, etc. can be discovered the same way they would if DHCP was on the LAN, assuming the devices were assigned IP addresses in the LAN network.