In answer to the second part of your question, there is a paper by Pleasants [[Zbl 0328.12008](http://www.zbmath.org/?q=an:0328.12008)],``[The number of generators of the integers a number field](https://doi.org/10.1112/S0025579300008548)'' in which it is shown that the number of generators is at most $\lceil \log_2(n) \rceil$, with equality when $2$ splits completely in the field.