Universal Turing Machine for every model has finite number of states, but it can simulate any Turing Machine with arbitrary large number of states. This is because all the state information of the simulated machine (actually the whole configuration) is kept into the tape.
So it is not the case of how finite states can hold potentially very large information. It is rather how (finite state + infinite tape) holds (very large number of states + infinite tape). Basically there is no problem with storing $(n^2 + \infty)$ information in $(n+\infty)$ space.
And there are two ways to enumerate Turing Machines. Easier way out is just take binary representation of $i$ as Turing Machine $M_i$. If it is a valid representation it is okay, if it is not, we shall assume it to be a Turing Machine that does not accept any input. The Universal Turing Machine will also honor this fact.
But if we are given a fixed representation and need to generate $i$'th valid Turing machine then we need to do it in hard way. We need to generate $\alpha_{M_i}$ the representation of $M_i$ one by one after checking if it is a valid Turing Machine. In this case the Universal Turing Machine will always get a valid representation of a Turing Machine.