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In HuffingtonPost's Facebook Page account, the Wall tab shows all posts.
But in my user Account, the wall tab does not show all post or tweets for anonymous.
It only shows for a logged user (Facebook).
How to show the wall that contains the posts of all users, even if they are not logged in?
My privacy settings are set for Everyone.
Are you referring to your News Feed (Top News), News Feed (Most Recent), or the Recent Activity on your profile?
I think that News Feed (Top News) only displays what Facebook considers to be most interesting to you, so I wouldn't expect your own likes/comments there. The News Feed setting that you mentioned applies to the Most Recent setting.
News Feed (Most Recent) does not display individual stories for comments and likes on posts; you would only see them under the corresponding post, if it is expanded. Are you saying that you expanded the likes/comments and you can see comments from other people but your comment is not there? Sometimes it takes some time for it to show up although it shouldn't take days (unless it was deleted).
If you are referring to the Recent Activity on your profile, there is an additional setting for that that you didn't mention. On your profile's Wall (not your News Feed), scroll to the bottom and click Edit Options. If you have hidden any kinds of stories from your Recent Activity these will be listed and you can unhide any of them by clicking the X and then Save.
As for the Photos tab, that used to be a known issue although the link to Facebook Help in that answer does not seem to work now. I don't know if that means that it should be fixed.
The Twitter updates go to your wall at the default level of privacy for wall posts. I think the people who created the Twitter app are responsible for adding the granularity for the privacy.
However, it doesn't matter, because the Twitter app cannot read your Twitter updates and repost them to Facebook unless your Twitter posts are public in the first place - public as in anyone can see them.
Best Answer
The HuffingtonPost's account has an additional "Fan Page" configured, whereas you are a regular user. Facebook makes a distinction and therefore has different access options.
See this Facebook FAQ link.
You can create one for yourself it seems, if you like via this create page.
Quoting from the FAQ: