One cannot say 'to better his English?' I use this construction all the time.
I've only heard this in a formal context but it seems fine to me. You should report it.
Yes. I'm a native speaker and I use that construction all the time, too. "To better" is a perfectly good verb, though it's not used as commonly as other verbs such as "to improve".
Duolingo should allow "ameliorate" as it is the direct, formal translation
migliore can also be a verb. as in to better oneself. Word Reference lists migliore as a verb.