after rooting phone, can I remove all traces of original phone carrier?
As its hardware is still altered by Verizon, you'll likely not be able to perfectly swap out the OS, since the bootloader of your device is still locked down by Verizon, and that makes sure you don't touch anything related to boot (it won't boot up if you do), so it is also impossible to replace the splash screen that shows up first when you boot the device up.

That being said, it still shouldn't complain about anything Verizon wants the device to complain about. Everything you see after the device ends the boot animation and displays the lock screen are controlled by the custom ROM. So if the custom ROM developer doesn't want the ROM to complain about anything (and that is kind of the point of a custom ROM!) you should be fine. Also Verizon would not be able to interfere with your device functions remotely.