In general, does a terminal degree such as a PhD from a major state university give one more options (i.e., more competitive programs) when applying for residency or fellowship? Is it possible for it to decrease one's competitiveness?

A PhD will, all other things being equal, make you more competitive. However, I would strongly advise you not to do a PhD just to increase your competitiveness for residency or fellowship. Do it only if you really like research. Otherwise, you will be miserable during your PhD.