Regarding the question around creating issues for your neighbour, it will depend heavily on how the DMZ feature is implemented on your neighbour's network.
One classical DMZ model uses public IP addresses, one per DMZ host, sometimes via NAT, sometimes directly.
Another approach uses private address space and port forwards, either forwarding all ports to one host with overrides where a different host requires a specific port.
In your case, on the (assumedly) residential router used by your neighbour, it's likely that it has a single public IP and DMZ mode exposes all ports (possibly filtered down to a smaller list) on that IP to a single internal address assigned to the "DMZ Host".
The impact of all this on your neighbour if you set the external address of your router as the DMZ host on their router is that they won't be able to do any port forwards to equipment on their internal network.
If their router is more sophisticated and allows for additional forwards to other hosts for specific ports, then it will only be an issue if you are wanting to forward the same port they that want.
You'll also need to consider the effects of UPNP here too, if your neighbour is running apps that instruct their router to temporarily listen on a given port and forward traffic back to them, it will either fail or will interfere with your forwards as your router won't be aware of these and likely will not be forwarding UPNP information on.
Another option available to you if DMZ is not possible, is to run a VPN service and forward just that port on your neighbour's router. Alternatively forward ssh, then tunnel everything else you'll require.