I was having the exact same problem as the OP and needed a solution with one slight difference. I needed DataGridViewSelectionMode.FullRowSelect not CellSelect. I was able to achieve this by doing the following:
datagridview1.SelectionMode = DataGridViewSelectionMode.FullRowSelect;
datagridview1.ClipboardCopyMode = DataGridViewClipboardCopyMode.Disable;
datagridview1.MultiSelect = false;
I believe setting ClipboardCopyMode disable defeats the built-in grid function and what's left is standard MS Windows behavior. So I am able to select text in an editable cell and enter ctrl-C to copy "just" the selected text.
-OR- It's quite possible there has been a behavior change between .NET versions since this was first posted. In any case this works for me using Visual Studio 2013, .NET 4.5 and windows 7.
Also important to note: I my case I only cared about being able to copy/paste editable columns/cells. This solution achieves this. Right click still works too.
Other grid settings that might be important to viewers:
datagridview1.EditMode = DataGridViewEditMode.EditProgrammatically;
datagridview1.EnableHeadersVisualStyles = false;
datagridview1.RowHeadersVisible = false;