Marchioness
ENVELOPE: "The Most Hon the Marchioness of ...." or socially, "The Marchioness of ...."
OPEN: "Madam" or socially "Dear Lady ..."
CLOSE: "Yours faithfully" or "Yours truly"

Marquess
ENVELOPE: "The Most Hon the Marquess of ...." or socially "The Marquess of ...."
OPEN: "My Lord," or socially, "Dear Lord,"
CLOSE: "Yours faithfully" or "Yours truly"

Mayor (men and women)
ENVELOPE: "The Right Worshipful the Major of ...." if a Mayor of a City; "The Worshipful the Mayor of ....." if a Mayor of a Borough or Town Mayor.
OPEN: "Sir" or "Madam" alternatively, "Dear Mr Mayor" can be used for a man or woman.
CLOSE: "Yours faithfully"

Members of Parliament
ENVELOPE: "[Edward Smith] Esq, MP"
OPEN: Dear Mr. [Smith]"
CLOSE: "Yours sincerely"

Members of Parliament - Cabinet Members
ENVELOPE: "The Rt Hon [Edward] ....MP", if a Privy Counsellor, as most are. If not, then "[Edward Smith] Esq, MP". Include "Sir" before first name if applicable.
OPEN: "Sir" or where the matter concerns the Department, "Dear Secretary of State," or "Dear Minister,"
CLOSE: "Yours faithfully"

Member of Parliament - Prime Minister
ENVELOPE: "The Rt Hon [Edward] ....MP". Include "Sir" before first name if applicable.
OPEN: "Sir" or "Dear Prime Minister,"
CLOSE: "Yours faithfully"

Military Officers

ENVELOPE: Professional rank should always precede any titles, eg. Adm [the Right Hon the Earl of .....]". Note, Lieutenants in the Army, Flying Officers, and Pilot Officers are addressed by their social title, eg [Edward Smith], Esq., RAF."
OPEN: Use their social title.
CLOSE: "Yours faithfully"

Prince
ENVELOPE: "His Royal Highness The Prince [Edward]" for the son of the Sovereign. "His Royal Highness Prince [Edward of ....]", for other Princes. "His Royal Highness The Duke of ...." if the Prince is also a Duke. Socially "Your Royal Highness, "
OPEN: "Sir"
CLOSE: "I have the honour to be, Sir. Your Royal Highness's most humble and obedient servant"

Princess
ENVELOPE: "Her Royal Highness The Princess Royal" for the daughter of the Sovereign. Her Royal Highness Princess [Jane] ....], for other Princesses. "Her Royal Highness The Duchess of ...." if also a Duchess. Socially "Your Royal Highness,"
OPEN: "Ma'am"
CLOSE: "I have the honour to be, Madam, Your Royal Highness's most humble and obedient servant"