If your router is configured to put your PC in a DMZ or perform port-forwarding, then presumably you can do this.
There are certainly folks that run Fiddler on Windows/Linux instances in the cloud (e.g. Azure).
If websites could uniquely identify devices, we wouldn't need cookies. Of course, that'd be a privacy/security nightmare. You can only get a remote IP's MAC address if you're directly connected to it, which is rarely the case in a given internet scenario.
If the client device cooperates, you could have your proxy return a HTTP/407 auth challenge and then the device would supply its ID as the proxy authentication response, but many devices and applications don't work well with HTTP/407s.
Check out the Telerik Platform - the only platform that combines a rich set of UI tools with powerful cloud services to develop web, hybrid and native mobile apps.