I recently purchased and installed a template from templatemonster.com to my Zen Cart site. All was going good until godaddy installed my SSL. Some elements on the home page of my site do not display in HTTPS but they do display in HTTP.
That sounds like the problem is with the cert and your preference for non-www’d URLs. The cert is provided for either www or non-www but not both so ask GoDaddy to use your preference (as configured in your ZenCart config files).
I don’t know how GoDaddy installs https but, if it’s in a different directory (not shared with the non-secure website), that might be another source of problems. These days, very few hosts do that, though.
While you’re at it, I’d also recommend using a good host rather than a registrar for your hosting. GoDaddy has a terrible reputation as a host while their reputation as a registrar is pretty good. Of course, if they fix the current problem, then the hosting choice is up to you.
Also: you don’t need https for the normal site. What do you think it will be stolen from there content? You don’t have to be hi tech to copy - paste.
My suggestion: use https only for the shopping chart. So only for the check-out pages.