Please allow me to answer your second question first: Chemical symbols, whether comprised of a single letter or of multiple letters, should be enclosed in Nemeth Code switch indicators wherever doing so would PREVENT CONFUSION. However, there is currently no requirement beyond this that chemical symbols outside of an equation or other mathematical expression be enclosed in Nemeth Code switch indicators.
My answer to your first question (whether "names like 'Copper(I), Iron(III), Cobalt(II), etc.' need to be enclosed in Nemeth Code indicators) is no, they do not. In narrative text, I would transcribe these names of compounds that contain no superscripts, symbols or operation or comparison, or other "technical material" in UEB, without Nemeth Code switch indicators.
Again, thank you for your questions, and thank you for your time.