after rooting phone, can I remove all traces of original phone carrier? Forgive my mobile ignorance.  I have a Galaxy S4 from Verizon (sch-i545), and I do not use it as a phone anymore.  What I want to do is turn it into basically a very simple media device for kids' use on wifi at home.  Almost a tablet.  I want a very clean version of Android, and I will install a very few number apps, just a few carefully-selected things that my children can use.

If I root the phone and then install a custom ROM (I haven't done this yet but see plenty of tutorials online), will I be able to achieve what I want, or will my phone keep complaining about not having a SIM card?  And will it still think it is associated with Verizon (custom splash screen and who-knows-what-else)?

Certainly, I could just factory reset it and disable the dozens of bloatware apps, but I'd feel especially content to do a little more than that and have the knowledge that it's just a little computer running  the custom android version I want without any Verizon fingerprints upon it.
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.