If your host has attempted to unilaterally upgrade your site, then run like hell. That can only end in tears and they shouldn't be doing it.
However, the sudden appearance of a new admin section and a Zen Cart trying to find the cache table using the unrecognized (but default) zen_ prefix, does sound like somebody has been playing with the site.
Although you can upgrade by installing 1.3.7, reimporting and then upgrading your database and copying across stylesheets and other template files that you have changed, you would also need to amend those stylesheets and other template files to reflect changes that have been made in the versions of Zen Cart that come later than that which you were running when you created them - so it's not trivial.
I wouldn't even start such an upgrade until I was clear about the changes had occurred in the first place and was confident that they wouldn't happen again.



