I have created a custom template for the category view that I am assigning to the LAST category in a tier. At the bottom of this template, I want to show all other sibling categories within the parent's category. This would be very similar to how the category shows the name, image, and description of its child categories, only I want the SIBLING categories, NOT CHILD.

How can I go about showing these other sibling categories on a category page?

A custom mod_menu or a K2 Tools module (category listing) would be an appropriate solution.

Is there a way to do this without using a module? Or, can I call a "universal" module from the category.php template that updates its contents based on the current category page loaded?

The quickest method would be the module. You would otherwise have to query the database, load the results and display the results after you find each category's route.

So, can I call a "universal" module from the category.php template that updates its contents based on the current category page loaded?

It is possible. This has to be custom PHP code though.

I'm already creating custom code for the K2 template. Can you give more info on how to achieve this?

You have to manually query the db and depending on the ID of the category you have to display the categories where the "parent" cell matches the ID of that category.