Do children still ever write actual paper letters and send them in the post?

Do adults?

Yes, some do. Mostly thank you letters to grandparents and stuff line that

I worked at a summer program for high school students about 2 years ago, and we had them write postcards that would be mailed to next year's program participants. Quite a few of them were really confused about the concept of a postcard. "We're just going to write on the back of it?", "How is it going to get there?", "But people can read what I wrote!"

I didn't anticipate this at all, and it was pretty amusing how put off people were by the "but someone can just read it" aspect. I guess kids care about privacy to some degree still.

Yes.