LTE networks require a UICC (SIM card) to authenticate and connect to them in the same manner that traditional GSM networks do. So for the Sprint HTC One with LTE, the SIM card is for Sprint's LTE. As for the other questions about using a SIM from another network, I can't say (though I know others will jump in here with the answer). It would require that the Sprint version be compatible with the other carriers' networks.
I just get sick of setting people call other people liars about their phones being for Sprint when half of them don't know what they are talking about lol. I carry Sprint I have a SIM and my daughter loves my phone and wants one I have plenty of credits to bid