How to finish proof that $T$ has an infinite model?

You are close.

Yes, this is what you need to do, but remember that inequality is not transitive. It is not enough to require $v_i\neq v_{i+1}$, but you need to have $\varphi_n=\bigwedge_{i\neq j<n} v_i\neq v_j$.

If there are arbitrarily large finite models, then every finite collection of sentences of the form above is consistent with $T$, therefore $T\cup\{\varphi_n\mid n\in\omega\}$ is consistent and any model of that cannot be finite.

This is a standard fact. The result you are looking for is exactly the compactness theorem, but you can also do it directly. Just take an ultraproduct of a sequence $M_i$ of larger and larger finite models. Since every one of these models $T$, so does the ultraproduct, by Łoś's theorem.