Full Circle

place the search widget in the header widget zone. this will give what you are after.

there are 2 files to use in the header of custom community
one is the header background image (non clickable) and logo.
the logo will give you a link to the home page when set.

to remove the menu from its default location
div#access{display:none;} in your css tab of the theme would remove it with out editing the code. as the theme update would remove your edits. the css will not.

to remove the default CC logos on custom community just upload your own header back ground image. in theme settings / header / header background image.

if you wish to not have a header background image. use a clear pixel gif image. set the background to repeate.

please remove all the code edits from the above comments as other functions of your theme will fail to work.

your old site is down so i can not see what you had.

can you post a link?

you can fix this by editing the page, right hand side set the page template to right and left sidebars.
this will over ride the default left only setting for only that page.

child theme and edits to the tag line is the only way to do this.