I am not an expert and do not own an Oppo 103, but thought I add a few things I do know. I think you are running into network/firewall/routing issues.
First, I would get the PC and the Oppo working together, then worry about Gizmo. Gizmo should be fine if you can get the PC and the Oppo to communicate. You want to use JRiver as a DNLA server and the Oppo as a DNLA renderer. First make sure you have the DLNA options checked in Tools/Options/Media Network/Advanced. If those are checked, the Oppo should show up in Playing Now. If it does not, it may be blocked by your firewall or you router. You can turn off your firewall and see if that helps. If it does, then you are going to have to figure out how to get through the firewall. If not, I am probably not much help, unfortunately. Hopefully, turning on the DLNA settings will be enough. I believe you have to have Media Center up and running on the PC. Just running the MC server itself may not be enough.
If your TV or your receiver are DLNA devices you can turn them on and see if you can see them in Playing Now in JRiver. I routinely use Gizmo to show the pictures on my JRiver PC through DLNA to my TV. If your receiver has DLNA, you might just bypass the Oppo and go directly to that. If either of those devices is a DLNA renderer and you do not see them in JRiver then the problem is broader than just the Oppo. Many new TVs and receivers are DLNA devices. If you can see one of those devices, you should be able to use Gizmo as you want to.
Unfortunately, if JRiver does not automatically see a DLNA device, it sometimes takes a bit to figure out the networking issue.
If your PC and Oppo are close to one another and your PC has HDMI out, you might also try connecting a HDMI cable between the PC and the Oppo and see if that shows up in Audio as an output device.
As to playing music to your phone outside the home, you need to set up port forwarding on you router so that when you connect to the router from outside you local network. You need to figure out the IP address that JRiver is using, go into your router/firewall and set port forwarding on the IP address to forward the link to the ip adress:52199. That setting is shown in the Media Network menu. Setting up port forward is different for each router. You need to log in as administrator and find the settings. If you are going to do this regularly, you should probably set up your JRiver PC as a fixed IP address. Otherwise, if the router changes it on reboot you will not be able to connect. You can do that in the Windows Control panel under networking. It should be something like 192.168.1.xx where you set xx to something not being used, like 50.
Sorry I do not have a quick answer, but this should at least give you some things to try.