Learning music without practicing it is permissible, but such a thing is merely a theoretical assumption since learning music is associated with listening to it and playing it.
It is prohibited to learn or play music that causes rapture.
It is permissible to delay the prayer after the beginning of its time, although it is better to perform it at the beginning of its time. The time for the Dhuhr and Asr prayers is from noon time when the sun starts declining until sunset and it is not permissible to delay it after sunset.
We cannot determine whether the said length of hair suffices for it to be considered a beard in the conventional sense. It is convention that will determine what a beard is. Apparently, the trimming undertaken at the lowest level of the conventional beard trimmer does not satisfy the minimum length of a beard.
Yes, it is permissible to do so during the waiting period (the Iddah). If the husband agrees, they return to their bond of marriage. After the Iddah ends, she is not allowed to seek the return of the payment. However, they can get married again with a new marriage contract.