I have observed that on Makeuseof, there is floating search bar with ask and search box. I would like to create one like it.
When user click on ask button they should be redirected to answer script installed in sub folder.
Try this one: http://allbloggingtips.com/2012/05/31/how-to-add-floating-notification-bar-in-wordpress/
Or this one: http://forums.themeshaper.com/topic/floating-the-search-box-in-the-header
Just so you know, I personally can't stand it when a site forces me to take screen room for things like this. I have a large screen so I can see plenty of things. Not so people can fit more advertizements or things onto it.
So, if you do this, my suggestion would be to keep it small. Or not at all.