Using the OBJ format has it limitations, like it won’t export the Rigging…Far better to use FBX and there is a section on the exporter for collecting textures to a file…
The eyes have several different parts as you have seen…I tend to use my own setup instead of the default, and I don’t use a glass shader like others will use ( I work almost exclusively in EeVee, unless I need displacement or Bevel node…)
This is my set-up for the eye cornea and moisture…which will revel the irises…and add in the specular reflections…
If you go the route with the Diffeomorphic add-on, then you can, with a bit of fiddling, just import the poses from an FBX file and apply them to a single exported character and you can also bake the textures into the FBX, using the export dialogue ( not to mention all the facial and body morphs you want!) But you should go over and read the section on the addon on BA