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)

Yes, I'm using MAMP. I'll do a backup with phpMyadmin. But what do I click to get an automatic upgrade?

You can try using the wordpress theme upgrade option, but you may have to give wordpress FTP access to your machine and I never do that. I do a manual upgrade.

As I explained in my message above (with attachment), when I click on the new version number, I don't arrive at any place where I can request the automatic upgrade. Can you help with that?

Looks like you need to manually upgrade

In FAQ (2009): There are two ways to update Atahualpa to a newer version
1) use the WordPress automatic update or
2) Manually update the theme

Why is the "automatic update" even mentioned, since it does not actually seem to exist? If it does exist, please tell me how I can do it.

The automatic update should be available in Appearance > Themes under the theme in question or in the Updates section of the WP Dashboard.

You are using a local site, the automatic upgrade works fine when your site is on a server at a host.
You can make it work in MAMP, you just have to setup MAMP correctly, something that is beyond the scope of this forum.

The attachment view looks exactly the same on my live, hosted site. The link there (on the phrase "Atahualpa 3.7.11 released" is http://wordpress.bytesforall.com/, which gives download information but no info on automatic update. Also on my live site, I attach shots of the Appearance > Themes under the theme in question and the Updates section of the WP Dashboard -- neither gives any clue how to update automatically.

if your atahualpa folder in the themes folder is not called 'atahualpa' then wordpress won't recognize it and look for an update.

My folder is called atahualpa353 -- I'm assuming it was originally installed that way; I did not rename it. Is this improper? Should I change it? How can I do that?

You can use FTP to rename it

You have not answered my other questions. Why was it installed with this name, if another name is required to upgrade it?

I tried simply renaming it to atahualpa in my local site and the site stopped functioning, so it seems something else is required.

You have not answered my other questions. Why was it installed with this name, if another name is required to upgrade it?

I have no idea why you installed it this way.

I tried simply renaming it to atahualpa in my local site and the site stopped functioning, so it seems something else is required

not having access to your local setup, I can't give you any advice, sorry.

when you go to dashboard->appearances->themes what do you see?

See screenshot:

so when you renamed the folder, WordPress couldn't find the existing theme (since the folder was differend and it defaults back to twenty-twelve...but you have removed that theme so it has nothing to work with.

Have you ever done an automatic upgrade of wordpress?

yes. recently I took the local from 3.0.5 to 3.4.1 -- and then to 3. 5.1, then fell back to 3.4.1 when there were problems.

Well if you can automatically update Wordpress on your local, you should be able to update the theme once you have named the folder back to atahualpa

When I rename it, I get the following message on my Dashboard (in red):
ERROR: The themes directory is either empty or doesn’t exist. Please check your installation.

go back and read post #21