As always, post your Royal Questions & I will get to them as soon as possible.
Question 1: How do you display your cards that you receive from the Royal Family? Is framing them ok to do?
Answer: We went into this some in May 2017 Q&A #10. Framing them is okay, but you have to worry about fading, and ink possibly changing colours. I have my letters in binders.
Question 2: How about Sweden? Are they responding?
Answer: Sweden responds occasionally. In the few years I've written to them, I've only gotten 2 replies. You can see my Replies from the Sweden Royals here.
Question 3: Have you written to either the Bhutan or Japanese royal family? If so, how would you go about addressing them?
Answer: I have written to the Bhutan Royals, never got a reply. I don't think I've written to the Japan Royals before, and have never seen a reply from them.
If by "addressing them" you mean, what are their address? You can find the addresses for the different royal families here.
If you mean how would you call them in your letters:
The King & Queen of Bhutan are both HM (His or Her Majesty), the other Royal Family HRH (His or Her Royal Highness). So, there isn't anything different than other royals. You could address them anyway people might normally address that royal:
TM The King and Queen
TM King and Queen of Bhutan
King Jigme Khesar & Queen Jetsun
The Emperor and Empress of Japan are HIM (His or Her Imperial Majesty), the rest of the family HIH (His or Her Imperial Highness). There are a lot of different ways to address them:
Their Imperial Majesty
TIH The Emperor and Empress
TIH Emperor and Empress of Japam
Emperor Akihito and Empress Michiko
Question 3: Do you know anyone who has ever received a personal reply from the Duke or Duchess of Cambridge? I know sometimes people get them from Camilla.
Answer: I don't know of anyone who has gotten a personal reply from the Duke or Duchess of Cambridge. What Camilla does is very unique.