1. You mentioned the simple List Template. Is that the template your category is set to use?

2. What template are you looking at/editing/styling against?
/bvmodules/themes/categorytemplates/simple list/
or
/bvmodules/categorytemplates/simple list/

3. Do you want your change to only effect categories that use the Simple List Template?

Kyle,
Are you creating your site theme by 1. editing the Foundation4 Responsive theme directly, 2. editing a duplicate of the Foundation4 Responsive theme, or 3. starting from scratch?

If your active theme is Foundation4 Responsive or a duplicate, the site is using the theme's version of the Simple List category template /bvmodules/themes/YOURTHEMENAME/categorytemplates/simple list/, not the Simple List category template located at bvmodules/categorytemplates/simple list/.

The ability to override core files from within a theme is a new BV 2015 feature. You can read about it on the blog here: http://www.bvcommerce.com/blog/post/2015/03/05/New-Theme-Architecture-in-BVC-2015
The Foundation4 Responsive theme makes extensive use of this new feature.

In order to target the Search content block within your template's Content Column, add at least one class to your category template.

Let's say I add a class to the out most <div> of the Simple List template.

class="simple-list"

Now I'll have the hook I need to target anything within that class from my stylesheet.

If I needed to style the Search Box content block text input I'd add this to my stylesheet:

.simple-list .search input[type=text] {
color: red
}

Only Search Box content blocks on categories which use the Simple List Category Template will be affected.

I hope this helps.

Glad to help.

Is your theme responsive? I was concerned, if it is, that browser resolution changes may affect your column fix since typically, responsive column widths flex. What fits at higher resolutions may not fit at lower. Just something to consider.

To address you're new question:

Content blocks live in the /bvmodules/contentblocks/ directory. The Category Menu content block is one of many you'll find there. To change the markup of Category Menu you'll need to edit or override(preferred method) the content block's view.ascx file.

Location of the core Category Menu view.ascx file...
/bvmodules/contentblocks/category menu/view.ascx

In BV Commerce 2015 we have the ability to override content blocks from within the theme.
To override the markup of the Category Menu content block, add a new directory to your theme called "ContentBlocks" and copy and paste the directory of the content block you wish to override into it.

So the directory structure will look like this...

BVModules/Themes/YOURTHEMENAME/ContentBlocks/Category Menu/

Then you can edit the view.ascx file inside your theme and leave the core files alone.
For more info, read: Powerful Theme Architecture of BVC 2015


Am I correct to assume that you are editing an existing theme?
If that is indeed the case it may not be pertinent, however, I feel like I should point out that the newer category menu content block (Category Menu Plus) is all I ever use. Category Menu is really just there for legacy themes.

If you haven't already, read the blog post entitled "The Power of Content Columns" for some useful information about content columns.