Last I remember - (and i could be all wet) Apple Push is 'optional' in that it even if you cannot push messages thru Apple Push - the mobile device still will poll the Traveler server to pick up mail. And, of course, it's not used at all for Android.
You stated: ' the url traveler.companyname.com/traveler did not work with Wi-Fi from across the street. I could not remote desktop either to the server by using the IP address unless I was connected to the network. '
I wouldn't expect that you would be able to remote desktop from across the street, but if you cannot browse to the traveler server from autside your internal network, your phones will not be able to, either.
From a web browser, (at home, for example) try to browse to: https://traveler.companyname.com - or https:<ipaddress> - the external IP for your server. If you cannot connect from there that's what you need to work on....