If I purchase a SIM-Free mobile, does this literally mean that I can put my existing SIM into the new phone?
Do I assume that nothing else is different?
What you probably mean is an unlocked phone, that can take any network SIM card. A SIM free phone just means without a SIM card but could be locked to a provider.
It might. Mobiles tend to be locked to a network only if the service provider has provided the handset at a discount. If you bought it from elsewhere then it should be unlocked and you can put any SIM in it.