I am currently at 3.5.3 and have had security issues, so I should upgrade to a current level of Atahualpa. Can I go directly to some high level (3.7.11?), or do I need to go step by step through each intermediate version? If only one jump, should I read all the intervening notes for relevant issues and changes I need to make?

You should start by doing a database backup.
Next read this thread

Once you have done that you can install 3.7.11 and make the jump

I would like to try the automatic update first, but when I click the link (on the Atahualpa theme options page), I get an odd display with no further action possible. (See before and after shots attached.) Afterword: Apparently only the "after" is visible here, but perhaps you can get the idea...

have you done a database backup using phpMyAdmin or a backup plugin?

This is my local site (on my Mac), and I have backed up the whole Mac (Time Machine) and also duplicated the wordpress folder.

That's fine but you should use phpMyadmin to backup the database. The database is not part of the wordpress folders. Are you using MAMP by any chance?

(I also do lots of testing and development on my mac)