Q: So I have a tmobile iphone 5s and it's stuck on the white screen with black logo I tried restoring it with iTunes

Turn your device off and connect your cable to the computer, but not the device just yet. Start up iTunes. Now, hold down the home button on your device and plug it in to the cable - don't let go of the button until iTunes tells you it's detected a device in recovery mode. Now you can restore to factory settings.

Has the phone been unlocked officially? Has the phone been jailbroken in the past? If yes, these could be reasons for the behavior of the phone.

iPhone: About unlocking

Unauthorized modification of iOS can cause security vulnerabilities, instability, shortened battery life, and other issues

Yes, don't worry about that, it's just an estimate, as long as the software restores, that's the important part.