If we have multiple unique RSA signed messages produced from the same private key, can we always find new valid signatures?
For $t$ signatures, I think we can have at most $\sum_{i=2}^{t}\binom{t}{i}$ new signatures. Is this correct?

That's the number of new (message, signature) pairs with message of the form $\prod m_i^{k_i}\bmod n$ with $k_i\in\{0,1\}$ and $1&lt;\sum k_i$; that is, we combine at least two existing messages. BTW, that simplifies to $2^t-t-1$.
But we can just as well compute the signature for message $\prod m_i^{k_i}\bmod n$ for any integers $k_i$. As noted in the question, $k_i=-1$ or more generally negative works too. And even more generally, we can compute as many (message, signature) pairs as we see fit if we know $(n,e)$, for we can find the message matching any desired signature, by raising the signature to the power $e$ modulo $n$. Problem from an attacker's perspective is that the messages thus obtained are largely random-like.
Conclusion: textbook RSA signature can be secure only if the parties that verify are selective in the messages they accept, and the signers selective in the messages they sign. That's where message padding for RSA signature comes to play.


Are inverses necessary to produce new signatures?

No. That's one of the tricks that can be pulled to form new (message, signature) pairs with specific properties of the message in attacks against some RSA signature practices.