1) It is located in the skin .ascx file for whatever skin you are using. It looks like you downloaded a skin from somewhere - lets call it the "Gold" skin.  So your skin file should be located in YourWebsite/Portals/0/Skins/Gold/homepage(or whatever name of skin that page is using).ascx

The search bar is inserted in the skin using a skin object.

I don't have a skin in front of me right now but these are the two tags that you should remove off of memory.  The top one should be located at the top of the skin - this is telling the skin where the Search skin object is located.  The second tag is located inside of the HTML itself, where the search box was placed.

<%@ Register TagPrefix="dnn" TagName="Search" Src="~/Admin/Skins/Search.ascx" %>  <---- REMOVE

<dnn:LINKS runat="server" ID="dnnSearch"  /> <---- REMOVE

2) The CSS is located in the same folder as the above file, and is named skin.css.  If you need to figure out what CSS tags you need to modify, I would just deconstruct your webpage in a browser source view.  So if you wanted to figure out the CSS style the Privacy Statement is using in Internet Explorer or Firefox - just left click - View Source - Ctrl + F - search for Privacy Statement - and you will be able to see what "class" it is using.  You can also figure out CSS styles by looking at the skins .ascx file that you modified in step 1. Most skins in Dotnetnuke make use of a .Normal class for the main body text.

Hope this helps

If you want to put background colors on the modules themselves I would apply the styles to the containers. Containers basically surround all modules.  You can set a default container under your Site Settings - and the selections come from whatever files you have in /Portals/0/Containers/ folder.  There are also container examples in the default skin folder.  Here is what the code will look like:

<div>
      <dnn:ACTIONS runat="server" id="dnnACTIONS"/>
      <dnn:ICON runat="server" id="dnnICON" />
      <dnn:VISIBILITY runat="server" id="dnnVISIBILITY" visible="False"/></div>
      <div class="BrownContainer"  id="ContentPane" runat="server"></div>
      <div>
      <dnn:ACTIONBUTTON1 runat="server" id="dnnACTIONBUTTON1" CommandName="AddContent.Action" DisplayIcon="True" DisplayLink="True" />
      <dnn:ACTIONBUTTON2 runat="server" id="dnnACTIONBUTTON2" CommandName="SyndicateModule.Action" DisplayIcon="True" DisplayLink="False" />&nbsp;<dnn:ACTIONBUTTON3 runat="server" id="dnnACTIONBUTTON3" CommandName="PrintModule.Action" DisplayIcon="True" DisplayLink="False" />&nbsp;<dnn:ACTIONBUTTON4 runat="server" id="dnnACTIONBUTTON4" CommandName="ModuleSettings.Action" DisplayIcon="True" DisplayLink="False" />
      </div>

The div with the id="contentpane" is where you content is going to be placed, so you can add a CSS style to that div.  You can place that CSS style right into the skin.css file for your attributes.  You can also use a container.css file if you want, but I like keeping all my styles in one place.

In my opionion things can start looking kind of boxy if you put backgrounds on all your modules. Are you sure you want to do this and not just put a background on your whole content area?