Well, your setup is definitely not your 'standard' one modem router domestic network - so may take a bit of time to work out where the problem is. First of all, you seem to have a lot of Apple kit. Have you tested your system's internet access from non-Apple gear, other than the N-50, may be a Windows/Linux computer or at least an Android device? It may help in the investigation. You also need to make sure you investigate the possibilty of a faulty DHCP mechanism by double checking the N-50 with the correct static settings, including correct IP for the default gateway (ie the router with NAT switched on, that has the N-50 on its LAN), etc.
I'd advise piecewise process of elimination testing, so first start by making sure that your ISP itself is not preventing the N-50 from accessing the vtuner service. So disconnect the main Airport Extreme & the rest of the home network. Just connect the N-50 to the Freebox with NAT & DHCP switched on to create a simple 'mini' local network with internet access and test the internet radio (for both with N-50 using DHCP & static settings).
If the N-50 can now access internet radio, test to see if can still work connected to the Airport Extreme, by now attaching the AE to the Freebox in bridge mode (again for both DHCP & static N-50 settings). Hopefully this works, so will be the fall back position if there is no other way to get the N-50 to connect to the internet radio.
If everything is still ok and assuming the Freebox has a DMZ facility, place the Airport Extreme in NAT & DHCP mode & put its IP in the DMZ (will require some thought as to how to best group the IPs controlled by more than one DHCP). Attach the N-50 to the Airport Extreme & again test the internet radio as before - this will also test if the N-50 works through the Extreme's NAT firewall in isolation from the Freebox. If all ok, test if the rest of the network devices are working as before, including internet accessing devices.
If it works, then problem resolved! It was probably due to some sort of NAT type interference (similar to 'double NAT') by the Freebox when configured as modem only, possibly due to some sort of fault.
If it doesn't work then the problem is most likely Airport Extreme's firewall blocking required ports by the vtuner service, so you'll need to investigate further in this area, eg find out what ports are required, why the AE is blocking them, set up port forwarding, etc.