Must every who is a friend be able to read everything that comes in on my wall on Facebook?
If someone writes to me and I don't want what they say to stay on my wall for everyone else
To read, can I get rid of it any way?

I believe you are able to delete posts that others make on your page. If you hover over the post, you will see the option to delete it appear.

There are a lot of ways to control what gets posted, who can see it, and who can post.

The best resource is Facebook itself in this Help Area: