3. I'm afraid that is not possible without some kind of custom script. I would probably add a simple search icon there, and the search bar below that (into a container element), which I would set to invisible by default. Then when the user clicks on the icon, the search bar container would change it's visiblity. This is the easiest solution I can think of.
It will however require some coding. In case you need help, I can recommend wpkraken.io, it's a new service from some of the envato authors, so far nobody complained about them.
If you like my products, don't forget to rate them on codecanyon :)