A little beyond what I've done before, but I'm giving it a shot! When I use powershell or command prompt, I get "connection timed out" when using the IP of the miner. It's connected over Ethernet with ports 8333 and 3333 open. When I simply try to ping the IP address, it gives me request timed out with one "destination host unreachable".
Your device either has no Ethernet connection or you're using the wrong IP. You can hook up an HDMI monitor or check your DHCP server to see if the miner has an IP and if so, which one.
I was actually able to finally connect to the pool by plugging the ethernet directly into the modem's router that also has the ports forwarded (I was incorrectly typing the IP into the command prompt for the SSH previously). What would you recommend I check on an Asus router's settings that may be causing the issue there, but not when plugged directly into the modem's router?
Thanks again!
You should only need 8333 forwarded, 3333 is an outgoing port for connecting to the pool, nothing should be coming into your network on that port unless you run a pool.
Sounds like you might have a double NAT problem if your modem is providing NAT and your Asus router is doing the same the incoming packets will get lost along the way.
This is getting into trickier network configuration than I would like to troubleshoot on a forum, but you need to either switch the isp's router into bridge or modem only mode, or put the asus in the DMZ on the ISPs router, or remove the ISP router and replace it with the Asus (if the Asus has the capability to do that), or remove the ASUS from the equation altogether if it can't.
If you have everything locked down on the ISP modem/router then I don't see why you would need the Asus to do the same thing.