Dear Jeff,

When writing to the Holy Father or the Pope, whether Catholic or not, one should always show a respect for him and the office He holds. The answer to your question is, in no way, a doctrinal one. It is based on a proper respect and courtesy from a Christian viewpoint.

I think you will find this portion from WikiHow.com helpful:

How to Address the Pope

1. Address the Pope as Your Holiness. Another acceptable way to address the Pope in writing would be Most Holy Father.
   
   Note that on the envelope, you should address the Pope as His Holiness, ________ with the Pope's name in the blank. For instance, if writing to Pope Francis, the envelope would read, His Holiness, Pope Francis.
   
   
2. Maintain a respectful tone. Throughout the body of the letter, your tone should be polite and gracious. You do not need to write in flowery language but your language should be on par with how you would speak or be expected to speak inside a Catholic church.
   
   
   • Avoid swearing, slang, derogatory language, or any other forms of impolite speech.
   • Write out everything you need or want to say but keep in mind that the Pope is a busy man. Instead of dragging on and spending extensive space on flattery, it would be better for all parties concerned if you kept the formalities basic and went directly into the purpose of your letter.
     
     
3. Conclude the letter politely.
   
   
   • As a Roman Catholic, you should conclude the letter with a sentence along the lines of, I have the honor to profess myself with the most profound respect. Your Holiness' most obedient and humble servant, before signing your name.
   • If you are not a Catholic, you can change the closing to something along the lines of, With every good wish to Your Holiness, I am, Sincerely Yours, followed by your signature.
   • Something as simple as With every best wish. Sincerely yours, with a signature after it would also work for a non-Catholic writing to the Pope.
   • Regardless of the exact wording you choose, the level of respect you demonstrate should correspond to the minimum level of respect you should have for someone in the Pope's position, at the very least. Someone who does not follow Catholic teaching or who disagrees with the Pope should still recognize his position of authority and close in a respectful manner. Anyone who does follow Catholic teaching should demonstrate the respect expected from one addressing the leader of his or her faith on Earth.
     

Side note: Know the Vatican mailing address. If you plan to send your letter by traditional mail, you should write the address on the envelop as:

• His Holiness, Pope Francis / Apostolic Palace / 00120 Vatican City.
  
  Note that you should separate the address onto separate lines according to the placements of the slashes, /, as follows:

His Holiness, Pope Francis
Apostolic Palace
00120 Vatican City

• Other ways to write the same address include:
  • His Holiness, Pope Francis PP. / 00120 Via del Pellegrino / Citta del Vaticano
  • His Holiness Pope Francis / Apostolic Palace / Vatican City
  • His Holiness Pope Francis / Vatican City State, 00120

Do not write Italy on the envelop as the country. The Vatican is considered its own independent nation, completely separate from Italy.

To my knowledge, there is no known working e-mail for the Pope, though a fax may work as the image on this page testifies:

• Where Do I write?

You may find additional interesting information on:

• The United States Embassy to the Holy See
  

or

• the Catholic Education Resource Center: How to Address Church Officials
  

You said:

• Should I try to include a self-addressed, stamped envelope?

I'm not sure where to acquire the stamp.

Along with your letter to the Pope just include another empty envelope with your return address in the middle of the envelope. You can get stamps where we all get them: the local Post office.

Just ask the Postal clerk how much postage he or she would recommend placing on the return self-addressed envelope which is coming back to you from the Vatican.

The Holy Father is a very busy man, so don't get your hopes up to high for a reply. Based on what I have heard, when he has responded, it usually takes a few months.

I hope this helps,

Mike