Can I send a WAN and LAN connection over the same cable?

Solution 1:

If both of your switches are smartswitches, which means they support vlan's, you can configure the switchports as follows:

1st switch

port 1          access vlan 10
port 2 thru 7   access vlan 20
port 8          trunk vlan 10,20

2nd switch

port 1          trunk vlan 10,20
port 2 thru 4   access vlan 20
port 5          access vlan 10

Here's your edited diagram:

Solution 2:

Technically, yes this is possible (See this Wikipedia Article).

You would be limited to using a 100Mb connection rather than Gigabit, but typically WAN connection is going to be less than 100Mb anyway.

As the article mentions, this is not standards compliant, but 10/10 and 10/100 Ethernet only use 4 of the 8 wires found in a typical CAT5/5e/6 cable, so you would have to remove the existing RJ-45 connectors from your current cable and attach two connectors on each end using 4 wires for each connector. At that point you would have one connector that would be used for WAN and another used for LAN.

Personally, I wouldn't go this route. Its a very hacky solution and you may end up with interference if you have a lot of traffic on those lines. I'd just fork up the extra cash for a cheap router to extend your WiFi.