Show all tags related to a category?
You can use the relatedTo param on craft.tags like you can with any Element inside Craft.

All Elements inside Craft can be related to one another. There is a .relatedTo parameter available on all of the query elements like craft.entries and craft.tags.  

You will need to pass a category object into this relatedTo parameter. I have included the group param below but this can be omitted if you want to query all of your tags.

{% set tags = craft.tags.group( 'groupName' ).relatedTo( category ) %}

More info on using Relations in Craft: https://craftcms.com/docs/relations



UPDATE

The relatedTo param takes one of the following:


An AssetFileModel, CategoryModel, EntryModel, UserModel, or TagModel
object 
An element’s ID 
An array of element objects and/or IDs


To get a CategoryModel using a slug inside your template you can use:

{% set category = craft.categories.slug( 'theslug' ).first() %}

This will set category to a CategoryModel which you can then use inside the .relatedTo param in the above example.



UPDATE 2

If you've set category to have their own URL's then category inside the template you've set to use is a CategoryModel.

You then set entries to all entries related to the category. Finally, use these entries to get all related tags.

{% set entries = craft.entries.relatedTo( category ).find() %}
{% set tags = craft.tags.relatedTo( entries ).find() %}