You have to cosign something before you can spend your coins.
And who can start that kind of chain of signatures? Are there all coins assigned to the creator, or do you start from a completely empty chain? If all coins are held by the creator, then how it is better than Ripple? And if there are no coins, then who signs the first one?
Fake transactions are invalid, and nodes won't even propagate invalid transactions.
How to check, if a given transaction is fake or not? If there is "no blockchain", then there is "no persistent history of any kind". You can have one transaction, sending coins from Alice to Bob, and another transaction, sending them from Alice to Charlie. How do you know, which transaction is real, and which is fake, if both signatures are correct from ECDSA point of view? And if one group of cosigners can tell you one thing, and another group of cosigners can tell you the opposite thing, how do you know, who can be trusted? Remember: you rejected Proof of Work, so all signatures on all sides are equally good, which means, that nodes can be easily fooled.
Option 1 is to sign nothing.
Do you mean "signing a zero hash", "signing a one hash", or "not including any signature at all"? If you don't include any signature, then it is unknown, if someone decided to not sign anything, or if you simply missed that P2P message (and then, the system is not resistant to dropped messages). If you sign zero hash, then any coin can be moved anywhere. If you sign one hash (or anything constant), then it is not resistant to SIGHASH_SINGLE bug. So, how exactly "nothing" is signed? You take SHA-256 of an empty string?
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 are many dead altcoins. If you don't have any brilliant idea, then your altcoin will follow the typical path of pump and dump, where early adopters first get everything they can, then your coin will be listed on any exchange, and reach its ATH, and then it will dump to the bottom, when everyone will sell it for BTC.
In case of Proof of Work, there are limits, for example one block per 10 minutes. If you have just signatures, without any Proof of Work, then there are no rate limits. Which means, that nodes can be endlessly flooded, by a lot of people, sending everything between themselves. And because there are no fees in your system, the cost of spamming the chain is literally zero.
There's no inflation because coins are not generated, so the coinbase is necessarily premined.
In that case, why it is better than Ripple? And if it isn't, then why don't you just use Ripple instead?
By using a verifiable random function.
Which one exactly?
but there will always be a natural average of 1:1
I don't think it is the case. You can look at some statistics from Bitcoin or other chains, and see, that many times, you have multiple inputs and outputs. And of course, if your system will for example only allow a single input and output, and would form some kind of account-based system, then just multiple transactions would be used to achieve the same thing.
When a cosigner is nonresponsive, the sender must replace their address.
What if all of them are nonresponsive? And who decides, if someone was nonresponsive, or if it was an arbitrary decision of a given coin owner? If anyone can affect the randomness, just by changing the address, then people can do that endlessly, until they will get the cosigners they want, and your randomness could be easily bypassed by anyone in that case, and replaced by "address mining", just to pick the non-random cosigners.
Of course, as always, I encourage you to try your ideas, to write some code, to make some test version, and to test everything in practice. But as you can see, if you throw away some element, for example Proof of Work, then there are many consequences of doing just that, and I don't know, if you are aware of them.
Also, every system should have its constraints, and assumptions. For example: Proof of Work assumes, that the mining majority is in honest hands. In case of 51% attacks, the system works much worse than described in the whitepaper. So, what are your constraints? Having 51% of all cosigners?