It does seem odd that you can connect from a public network (e.g. your HTC device) and not from a PC browser. This is assuming your HTC is not connected to your local wireless lan as the Wowza server at the time. Please check out this article for basic settings for remote access to the Engine Manager. The important point in that article is if you're connecting to the manager remotely then you do not click the "+Server" and change the URL from http://localhost:8087 (though changing it to http://127.0.0.1:8087 can sometimes fix rare issues with localhost resolving).
I'd try some simple tests like telnetting to the port from an external network, but out of the box Wowza Engine Manager should accept connections from the Internet so long as the router is configured (port forwarding 1935 to local LAN address of server etc).
Regarding port 8086, this is the Admin interface port. It can provide useful stats in XML format, show the state of the Wowza instance, or even control certain aspects e.g.