If there are no fees, then what is the incentive to sign anything?
You have to cosign something before you can spend your coins.
that person can just sign fake transactions, or just reject to sign anything, and halt your network entirely, by making you unable to send any transaction at all.
Fake transactions are invalid, and nodes won't even propagate invalid transactions.
Option 1 is to sign nothing. When this happens, a sender can republish their transaction with a different address for the same UTXO. When that transaction is cosigned, the unsigned transaction is discarded.
Also, when your network will just start, then there will be just a few people interested in your system. And then, if you have for example 5 users, then it doesn't really matter, which one will be randomly picked.
There's at least one user per address, but there can be many addresses per user, so we can never know how many users there are. Only that the number of users is equal to or less than the number of addresses. In any case, it is highly unlikely for a new coin on an exchange to have such a small number of addresses for more than a fraction of a second.
There's no inflation because coins are not generated, so the coin supply is 100% premined.