As always, post your Royal Questions & I will get to them as soon as possible.
Question 1: Could we send letters to Her Majesty the Queen directly to Sandringham House? Or shall we always write to Buckingham Palace?
Answer: Always write to Buckingham Palace. The Queen isn't the one who answers her letters, it is the Correspondence Section at Buckingham Palace. So, you always want to write to Buckingham Palace. This way it will get to the correct people, whose job it is to deal with the Queen's mail. Letters sent to other royal residence will likely get forwarded to Buckingham Palace to be dealt with by the correspondence section.
It's a lot of work to process and sort the hundreds of pieces of mail the Queen receives everyday. You have to open it, figure out who it is really meant for. There will be some letters that should be forwarded to the Queen's Private Secretary, the Master of the Household, the Lord Chamberlain, or the head of the Royal Collection's office, etc. Harassing or threatening letter need to be forwarded to Scotland Yard. Then you would need to sort the mail for reason for writing, or event. And this would be all before one of the Queen's Ladies-in-Waiting even sees a letters.
Occasionally, you see people get responses from Ladies-in-Waiting from other royal residences: Sandringham, Windsor, etc. All those responses are from letters sent to Buckingham Palace, that were processes, and sorted there. Then some were taken along with the Ladies-in-Waiting, so they have some work to do during their stay at the royal residence. You just have to get lucky to get a reply from one of the royal residences.
Question 2: Does Her Majesty the Queen see "normal" letters from normal people? Is there any sort of selection from the ladies-in-waiting who would tell her about the most beautiful letters or something like that?
Answer: Yes, the Queen is shown a selection of letter from a diverse array of people and issues. We don't have a way of knowing which letters she read, as the replies are handled by ladies-in-waiting like the rest of the letters. But certainly if you get a reply from the Royal Residence the Queen was staying at, and it's a more personal reply, there is a chance that the Queen may have seen your letter.
Question 3: Do Kate and William sign autographs at public walkabouts?
Answer: No. The Royals are not allowed to sign autographs on walkabouts.
Question 4: Do you know when the Duchess of Cambridges birthday reply will be sent out?
Answer: They will send out replies when they get to it. They don't really have a set time they do it each year. When you write for the same event each year you will find the response time varies, it just depends how busy they are. Clarence House gets hundreds of replies a day, and correspondence section staff isn't that big.
But on average it takes about 1-2 months to get a reply. So expect something in February, but don't be surprised if it comes in early March.
I've gotten a reply in Mid January, before I even got a reply for my Christmas letter to the Cambridges. I've also gotten a reply in late February.