Removing The Middle of The Background, Keeping Only The Border

How do I remove the gray border that surrounds background images?

So, you have an img element that doesn't have a src attribute, but it does have a background-image style applied.

I'd say that the gray border is the 'placeholder' for where the image would be, if you'd specified a src attribute.

If you don't want a 'foreground' image, then don't use an img tag - you've already stated that changing to a div solves the problem, why not go with that solution?

Parts of the background are forming a border around the red border

Seems fine to me, what exactly is the issue? This effect you described might be either yous screen blurring pixels in certain zoom levels, or a visual illusion called Mach bands

Remove strip background keep panel border

If you set element_blank() for strip.background and keep element_rect(colour="black", fill = NA) for panel.border then top edge of panel.border will be black.
As pointed out by @adrien, for panel.background fill should be set to NA to avoid covering of points (already set as default for theme_bw()).

ggplot(mtcars, aes(mpg, hp)) + geom_point() +
  facet_wrap(~carb, ncol = 3) + theme_bw() +
  theme(panel.grid.major = element_blank(),
        panel.grid.minor = element_blank(),
        strip.background = element_blank(),
        panel.border = element_rect(colour = "black", fill = NA))

Sample Image

Remove border on only one side for end li (Maintain splitting borders)

Give it a class and then assign a no right border to that class. Check it out:

HTML:

<div id="order">
<ul>
    <li>Order</li>
    <li>Account</li>
    <li class="noRight">Login</li>
</ul>
</div>

CSS:

#order {
    font-family: HelveticaLTStd-Light;
    font-size:13;
    position: absolute;
    right:218;
    top:-5;
}

#order ul li {
    float:left;
    padding: 5px;
    border-right: 1px solid;
}

#order ul li.noRight {
    border-right:0;
}

Heres a JSFiddle for you.

How to remove the bottom border of a box with CSS

Just add in: border-bottom: none;

#index-03 {
    position:absolute;
    border: .1px solid #900;
    border-bottom: none;
    left:0px;
    top:102px;
    width:900px;
    height:27px;
}

How to remove ul unordered list's last li list item's border using CSS without changing anything in HTML code?

Add this style and you don't need to modify anything else:

#navlist li:last-child { border-right:0px; }

Edit:

Original Script

Original script to which the answer applies is posted here because jsbin.com may delete content not viewed for 3 months.

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
  "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
<title>Sandbox</title>
<meta http-equiv="Content-type" content="text/html; charset=utf-8" />
<style type="text/css" media="screen">
body { background-color: #fff; font: 16px Helvetica, Arial; color: #000; }
#navlist li
{
display: inline;
list-style-type: none;
padding:0 20px 0 20px;border-right:1px solid red;
}
/* !!!!!!!!!!!!!! PASTE ANSWER HERE TO MAKE THE FIX !!!!!!!!!!!!!!!! */
</style>
</head>
<body>
  <p>Hello from JS Bin</p>
  <p id="hello"></p>
  <ul id="navlist">
<li id="active"><a href="#" id="current">Item one</a></li>
<li><a href="#">Item two</a></li>
<li><a href="#">Item three</a></li>
<li><a href="#">Item four</a></li>
<li><a href="#">Item five</a></li>
</ul>
</body>
</html>