Attached Link: Examples

Thanks, I will be sure then to use only panoramic and/or longitude/latitude HDRI files.

In the pic, you'll find some of the renders.  You'll see that the hair object never failed on transparency which I thought was... perplexing.   As I said in my first post, all transparency boxes were checked on - both in the GI section and regular options section.

Image A - was what was produced after the first render that worked with transparency but rendered blue.  All the renders after that had no brow and lash transparency.   One of those is Image B.  To me, the lighting on the model doesn't seem to match considering the brightness of the light coming from the upper right in the back of the image.

Image C - must be one of those wrong HDRI files for Carrara as it (along with some others I tried) only produced a background image to the horizontal halfway point...

Image D - after rebooting and reimporting my poser file, I was able to get consistant brow and lash transparency.  However, I also rendered each new image to a new window.  Prior to that I had been rendering over bad renders.  I wonder if that had something to do with the transparency checkbox not reading itself properly... or something.   Also Image D uses the same HDRI file as Image A; it's been rotated 180 degrees though and seems to match better lighting wise.  (In fact, the brightness on the model I think must be coming from that bright right side in Image A which is now behind the camera on the left since it was rotated.

So my conclusions are to use the proper HDRI, and to always do new renders to a new window.