Hmm, it's hard to say what's causing the problem with the nav bar and the sidebar width. It's not that way on the demo site:
I would deactivate any plugins to ensure one of them is not causing the problem. Then revert any CSS changes one by one until the nav bar and widget look right again... that should help find the style change that broke it.
You can set the
background-color property on the
.widget-area div, like so:
background-color: #fff; /* Or whatever color you want the background to be */
We can't really help with CSS in this forum, but there are some resources here. I always recommend working in a child theme or with a CSS plugin (like Jetpack) to do this. You can also use a browser plugin like Firebug for CSS work--makes it much easier to pinpoint the areas you want to change.
I hope this helps -- good luck!