Do I need a minimum deposit (existential deposit) to keep my web3name live?
No. web3name is linked to your decentralized identifier (DID). A DID is not a wallet address, it is a digital identifier that is unique to you. Think of it as your fingerprint.
The w3n is your name, which you attach to this identifier. There is no minimum deposit needed for this.
The DID belongs to you as long as you remember the 12 words you used for creating it or until you actively delete it. You can use the DID to sign documents with, to attach credentials from various attesters to, or to associate crypto addresses from different blockchains with it.