Whether new parent menu was published?

Yes. The new parent menu item is published. But I can't put anything under it. I think something is wrong in the database.

Run this query in phpmyadmin

SELECT  `published` FROM  `#__menu` WHERE `alias` = 'root';

replace #__ with table prefix and show result.

it gives this:
Showing rows 0 - 0 (1 total, Query took 0.0006 seconds.)
SELECT published FROM wchs_menu WHERE alias = 'root'

The #__menu table does not have a very important record. Make a copy of the #__menu table and then do the following query

INSERT INTO `#__menu` (`id`, `menutype`, `title`, `alias`, `note`, `path`, `link`, `type`, `published`, `parent_id`, `level`, `component_id`, `checked_out`, `checked_out_time`, `browserNav`, `access`, `img`, `template_style_id`, `params`, `lft`, `rgt`, `home`, `language`, `client_id`) VALUES
(1, '', 'Menu_Item_Root', 'root', '', '', '', '', 1, 0, 0, 0, 0, '0000-00-00 00:00:00', 0, 0, '', 0, '', 0, 43, 0, '*', 0);

I don't think I understand. Do you mean export a copy? It seems to have a lot in the wchs_menu.

I tried running anyway and got error

#1062 - Duplicate entry '1' for key 'PRIMARY'

Do not do the above.

Be careful, create a database back up.
I suggest you rebuild the menu by clicking the "Rebuild" button on the menu page. Then try to move menu item again.

You can also unpublish your parent menu and republish again and check if does it work.

Ok new information.
Rebuilding alone doesn't work, nothing would publish. So I looked at the database again. Even though it shows in the backend as parent of 'students' the database is aiming at a non existent parent id. I moved my replacement parent item to that ID number and that fixed much.

However now I have the odd problem of the path going blooey. For instance, I move an item under a parent item called 'students'. In the backend it shows correct, but in the front it shows in the sub menu under 'home'. A look in the database shows the path has altered to home/whatever even though parent id is now correct.

At times I think I've got it, it looks fine and then I refresh the page and it reverts. I tried running rebuild menu and that works for awhile but it doesn't last.

Also getting this really odd message: Error
Save failed with the following error: MenusTableMenu::moveByReference() is trying to make record ID 105 a child of itself.

@Arydiel have you asked help on the forums as this repository concerns Joomla coding.

IMO this is not a Joomla issue.

Rebuilding alone doesn't work, nothing would publish.

That mean you probably have not run every sql queries on update.
You have a little mess in #__menu table.

Record (row) with alias = 'root' should have column published = 1, check this in your phpmyadmin.
Next go to admin menu list and click on search tools and change select status to all.
You may see menu items which are trashed. Publish them.

:( I have asked joomla forums for the original issue but didn't get a response. and menu item root is there and published. We already have deleted all trashed items.

At times I think I've got it, it looks fine and then I refresh the page and it reverts. I tried running rebuild menu and that works for awhile but it doesn't last.

Can you make a website copy, and unpublish all 3rd party system plugins ?

I am closing this as it is not a core code issue but is specific to your site