scrambler

You also need to adjust the margin left from 210 to 280px in the rule
.menu-add {margin-left:280px !important;}

If you really cannot find the rule, add the one below to cancel it
#wrap-header #nav-bottom-menu {z-index:auto;}

You still have the rules
#wrap-header #nav-bottom-menu {position:relative;z-index:95;}

Did you add that rule as I don’t believe it comes from weaver. It would either be in the Custom CSS Rule box, or one of the CSS+ boxes in the menu page

You need to remove that

#sidebar_header {overflow:visible;}
#sidebar_header .header-widget {padding-bottom:0;margin-bottom:0;}

yes and this is because you used the Horizontal header widget area. You must have read a post I made on the Weaver forum because you added
#sidebar_header {overflow:visible;}
#sidebar_header .header-widget {padding-bottom:0;margin-bottom:0;}

which is to remedy an overlay issue with that widget area, but because you are actually bringing your menu up over the widget area, it does not suffice.

The best would be to simply use the header image. You can use the same image and check the option in MAin Options > HEader > Header Image > Use actual image size
Just getting rid of the Header Horizontal Widget area and putting your image as the header image may be all you need, or we may need to re tweak the menu.

It will be easier to see once you have the header image in place

If what you mean is that you want the image in the Horizontal header widget area to be above the menu bar, the #branding has a z-index of 2 , and #nav-bottom-menu has one of 95, so you need to add the following to the Custom CSS Rule box

#branding {z-index:100;}

To clear that. I believe your z-index:500 rule has no use so you can get rid of it.

But this is not a responsive design. If you shrink your browser size, you will see the content of your menu bar slide under the image and disappear.

To avoid that problem, also add

.menu-add {margin-left:210px !important;}

Still not 100% clear 🙂

But When doing a search, the header image on the search page is the main header image of the site, not the one from the search results. The reason is that you may get more than one page in the search results.

So does the page in question uses the main site header image (the one for all pages), or does that page uses a different header image specific to that page. If so, it is normal that when searching the page, the search page result do not show that page header image as explained above.

Ultimately, we need a link to the pages in questions to be sure of what you mean and may be going on.

have a look at this tip page on the weaver forum, it gives you everything you need.
http://forum.weavertheme.com/discussion/2648/changing-the-background-image-for-a-specific-page#Item_5

In general you will get faster support on the weaver forum

You should explain what you are trying to achieve.

If you are trying to have a different header image on specific pages, Weaver has several mechanisms for that, so you may not need that plugin.

On any given page, selecting a featured image that is as wide as the theme will automatically replace the header image for that page.

If you want to select a different header image for specific page types, like certain category pages or other, you can use Weaver Filter box to do that as explained in the thread below.
http://forum.weavertheme.com/discussion/8003/infos-and-examples-for-custom-css-rules-part-10#Item_3
under Adding filters to change header image for different types of pages