For any skinning, including placement or removal of the topnav section, you should NOT modify the \App_MasterPages\layout.master. You should only modify the layout.master in your skin folder (\data\sites\1\skins\YourSkin).
In cases such as yours where you don't want a topnav section, I generally suggest that you include the following in your footer:
<portal:SiteMapLink id="SiteMapLink2" runat="server" CssClass="sitelink" RenderAsListItem="true" />
<portal:UserProfileLink id="UserProfileLink" runat="server" RenderAsListItem="true" />
<portal:LogoutLink id="LogoutLink" runat="server" RenderAsListItem="true" />
The RenderAsListItem value is up to you, it really depends on your site design. I suggest adding these items to the footer so 1) users can get to your SiteMap, 2) Admins can change their password easily or set other profile options and, 3) Admins can logout easily. The UserProfileLink and the LogoutLink will not display to those who are not logged in.
For your Search link, you can create a page and set the URL to http://YourWebSite.Url/SearchResults.aspx.
For the Search Filter box, add the following to your user.config and then open and save your web.config:
<add key="DisableSearchFeatureFilters" value="true" />
There really isn't an easy solution for hiding the feature you do not want to use without using Child Sites but I don't necessarily recommend using Child Sites for that reason alone. You can use the Feature Installation tool found in Administration > Advanced Tools to hide individual features from the available features dropdown. The only problem is, these settings will be overwritten when you install any mojoportal updates. You could remove the setup scripts for each feature that you do not want to use from the distribution package AND the site before upgrading your site. These scripts can be found in /Setup/applications.