By having enabled archives, I was indeed referring to having the "has archive" setting set to true in the Custom Post page.
I would have you do the following:
1) Trash the pages that you've made with the same names or URLs as your custom post types. They'll override the default archive URL and just show you a blank page, and you don't want that! There is a way to custom-code the loop into a template and set it up with pages, but that's an extra step and is kind of unnecessary because of WordPress's native functionality.
2) Double-check that the custom post types in question have "has archive" set to true.
3) Go into Settings > Permalinks and hit "Save" at the bottom. This will flush and rewrite the permalink structure to make sure they're directed to the right place.
4) Navigate to yoursite.com/posttype. Be sure to use the SINGULAR form of the post type (i.e. post, not posts). This should naturally draw on the archive template and show you a list of only that custom post type.
1) You don't need access to your server, but it can be helpful to have that in order to make certain changes. Developers are notoriously unwilling to give access to non-devs because they frequently don't understand processes and make mistakes that devs then have to clean up.
I would ask your dev for access and then ask him to lay out best practices for you, i.e. always backing up (always!), never making changes to the production site, not editing functions.php (or any PHP file, really) via the WordPress editor, etc.
2) As for the "get code" for each custom post type, that's an alternative method for adding it. You can either create/manage them via the custom post type interface/plugin, or you can take that automatically generated code and put it onto your functions file. If you're not that comfortable with editing code, I would recommend that you remove your additions to the function.php file and just stick with the plugin.
Hope this works out for you!