Your Royal Highness or Imperial Highness
It depends on the country and rank of the Princess because some princesses have the style "Highness" insetad of Royal or Imperial Highness.
Dear Respected Madam,
A correct way to address an envelope to Spain is: Senor (Mr.) Senora (Mrs.) or Senorita (Miss.)
There are two ways. Either as the username - @username - or as the full URL address.
sup hoe !
Dear Mufti Surname
put it up your bum
The formal, correct way to address a cardinal is "Your Eminence".
a quick web search - which could, of course, be used to answer most of the questions on this site - indicates that both princess's and princess' are correct possessive forms of "princess."
To The Hon. <name>
good evining master how can i hello sir
The Revd. & Mr. X
You can pretend you are a princess or you can marry someone who has the correct royal station to give you the title of Princess.