Ah, AP/Router firmware! Forgot to mention that. There are lots of ways to ping from the phone, but I don't think there is any software that is built-in. I've used ConnectBot, which is an SSH shell program designed to connect to SSH servers. I just connected to the local host, then pinged from the local host. This app may be a bit more than you need though, and I about need a magnifying glass to see the shell's font. Check Google Play for different apps, and maybe others can make some recommendations.
WiFi and your home ISP connections can be a pain to troubleshoot, and voice requires some serious bandwidth with very little room for latency or dropped packets. Both of those issues delay the voice packets, or worse, and humans don't accept someone's voice not sounding 95% perfect. With all that said, I'd hope that the software could warn the user that the connection isn't optimal when it notices latency or dropped packets instead of making them figure it out through a series of troubleshooting hoops. Before you send it back, call T-Mobile support and maybe they have some easier troubleshooting utilities or at least they can tell you if this is a known issue and may be fixed in an upcoming software release.