In particular, I want to know whether the smart contract is somehow its own entity in the blockchain, or if a smart contract is just a property that EUTxOs may pick up.
EUTxOs are bound by smart contracts via their script addresses. But where actually is the script? To make it more practical-- how does a dapp dev actually get his smart contracts on-chain for others to use?
Some wild ideas passing through my head...
a) you have to reference some off-chain script to cause your EUTxO to operate under it.
b) you have to use a reference EUTxO, whose only purpose is to give a generic source for the smart contract's script,
c) or else, maybe the script is deployed independently on-chain somehow (outside of an EUTxO).
I'm quite ignorant about how EUTxO works; so, a), b), and c) are just to help show you where you need to do surgery.