You can make emergency calls without a SIM card. The phone identifies itself to the network (with IMEI), even if there is no SIM. The telcos, obviously, map IMEIs to SIMs. (Perhaps needless to say, but for basic operational purposes they have to quite efficiently triangulate your position, to know which cell tower to instruct your phone to use.)

Having no SIM does not help you conceal your phone's location at all, only perhaps to make it harder to map it to your identity.

I don't know if any location data might leak in airplane mode, but I would not be surprised if some did, for example, through NFC or Bluetooth.