Hello,
If you look at the source code for your container you will see some code which is for an ACTIONBUTTON, depending on the container there maybe several action buttons (In DNN Blue there are 5), you just need to delete the code for the action button that you do not wish to display ie in a container.ascx file:
< dnn:ACTIONBUTTON4 runat="server" id="dnnACTIONBUTTON4" CommandName="ModuleSettings.Action" DisplayIcon="True" DisplayLink="False" / >
For the search, yes you can go into the skin code and remove the [search] token in a .htm skin - or for the DNN-Blue skin look at the .ascx file and remove:
< dnn:SEARCH runat="server" id="dnnSEARCH" / >
If you are just getting started with editing skins, it will be worth your time checking out the tutorials in
Issue no 5 & 6 as it provides the beginning foundations for getting started with skins in DNN.