Meghan Markle is marrying into one of world's oldest monarchies, and while she will likely enjoy the title "Her Royal Highness" and all the perks that come with it, she will not be a princess.
The explanation is a bit complicated; only those born into royalty have the privilege of prince or princess preceding their name. Case in point: Prince George, Princess Charlotte and Princess Eugenie, to name a few. Each is a grandchild or great-grandchild of the monarch and therefore is style HRH Prince(ss) ___.
To call the Duchess of Cambridge "Princess Kate/Catherine" or even "Duchess Kate/Catherine" is incorrect since the former Kate Middleton was born a commoner. The same rule applied to the late Princess of Wales, who, even though she was born into an aristocratic family, is not royal by blood. So Princess Diana, although widely used, isn't correct.
To make things even more confusing, women who marry into the royal family take their husband's rank. For example, Catherine is "Princess William of the United Kingdom," but she is not "Princess Catherine of the United Kingdom. The same will apply to Markle once she weds Harry.
The Queen is the only person who can gift titles to couples getting married, and usually does so in the days leading up to their wedding.
As for curtsying, after Catherine and William married, the rules were refreshed as a reminder of royal rank. For example, if William is present and because he is second-in-line to the throne, Catherine is only expected to curtsy to the Queen, Prince Philip and Prince Charles.
If Catherine is alone, she must curtsy to everyone in the royal family, since they all outrank her. The same rules will apply to Markle, except she curtsies to William and Catherine. In truth, William and Catherine will likely insist that Markle not curtsy to them. Camilla has reportedly told Catherine not to curtsy to her, since the two women enjoy a close relationship.
Got all that?