"Is there a module that I can display, such as latest media, via a command similar to {loadmodule mod_hotornotfactory_items} ?
There is no such feature implemented.
"Also, when you say to update the module layout code with code from another layout, can you tell me which layout to copy from, and which to copy to?"
components/com_hotornotfactory/views/items/tmpl/default.php to modules/mod_hotornotfactory_items/tmpl/default.php
Take note, that other changes might be required to make this work. Php and Joomla knowledte is required.
"I do not see a module that specifically seems to be displaying latest media - I would expect it to be called "mod_hotornotfactory_latestitems" or similar."
To display latest media, use the mod_hotornotfactory_items module and select Latest Added from the module's Display items option.
More info on how to use the loadmodule option here:
docs.joomla.org/How_do_you_put_a_module_inside_an_article%3F