Creating custom profile fields with a plugin
 

Can custom profile fields be created with a plugin? If so, how, and how can the fields be accessed since the field number is not known before installation?

Yes they can. Try a search in the mods area and find a modification that has done this. I think there is one that does country flags and another that does gender. So, search it out, download the plugin code, and see how they did it.

I looked at all the top mods for these two keywords and did not find any that created custom fields upon installation. I looked at:

https://vborg.vbsupport.ru/showthread.php?t=117973 no php in this plugin
https://vborg.vbsupport.ru/showthread.php?t=196749 field must be added manually
https://vborg.vbsupport.ru/showthread.php?t=231260 field must be added manually

https://vborg.vbsupport.ru/showthread.php?t=238863 field must be added manually
https://vborg.vbsupport.ru/showthread.php?t=185841 field must be added manually
https://vborg.vbsupport.ru/showthread.php?t=229511 field must be added manually
https://vborg.vbsupport.ru/showthread.php?t=155223 field must be added manually
https://vborg.vbsupport.ru/showthread.php?t=159252 field must be added manually

This is something I plan on doing with one of my modifications, as well. But I haven't yet looked into it. I have, however, bookmarked this thread for when I'm ready to try it out: Plugin How To - How to add custom field to existing Vbulletin table

Thanks for that link. I'll look into it.