When Charles becomes King, will Princess Ann keep the Style Princess Royal?
Yes. Princess Mary (Queen Elizabeth's aunt) kept the style for life even after her brothers became kings.
One of Duchess’s Kate’s styles is “HRH Princess William” so does that apply to males that marry into the Royals? Should Prince Philip also be known as HRH King Elizabeth II?
In the UK, women take the titles and styles of their husbands, but the reverse is not true, i.e. husbands don't take the titles and styles of their wives.
For example, Princess Margaret became HRH The Princess Margaret, Countess of Snowdon when her husband was created Earl of Snowdon, but her husband was never styled HRH The Prince Margaret. In order to become an HRH, the husband of a British princess has to be given that style in his own right as was the case with Prince Philip.
Note that other countries might follow different rules. For example, in Spain, the husbands of women who hold titles of nobility in their own right also use their wives' titles, e.g. the husband of the Duchess of Alba would be called the Duke of Alba. That used to be true also for Spanish royal titles. In fact, even the husband of a reigning queen like Queen Isabella II for example was called HM The King. Nowadays, maybe due to northern European influence, that is no longer the case as, according to a royal decree issued by King Juan Carlos in 1987, the husband of future reigning queens will have the dignity only of "Prince" with the style HRH. Likewise, husbands of "infantas" (Spanish princesses who are not the heir to the throne) are no longer "infantes" or HRHs, but, if their wives are also duchesses, they can still use their wives' ducal titles by courtesy with the style of "Excellency" as customary for Spanish dukes. The husband of the Princess of Asturias (the female heir to the Spanish throne) is the only royal husband who can still use all of his wife's titles and styles (including "HRH The Prince of Asturias") as in the traditional Spanish custom.