I added a couple of new tags (System > Settings > Posting > Tags)
but I can't seem to move them "up" so that the tag menu is alphabetized.
Where can I go to do this? Is there some *.ini or Settings file which contains the tags?
(I have tried to look for such a file, but not found so far.)

Also, the cursor cannot be placed into the Defined Tags display anywhere but at the end.
Free cursor placement would enable adding a tag alphabetically.

This would require customization if you would like to do this at present. 

You can of course, provide your suggestion in our official Feature Suggestion forum for further evaluation.

Customization is OK. Where do I go, please?

Sorry customizations are outside our scope of support. You can contact the following developers at the link here for their services or post in the Developer Assistance forum.

You can sort them alphabetically with this plugin.