Can I create Template for archive listing pages (category, tags, search)?

You can create a special template for archive listing pages (categories, tags, author posts listing, search results). 

Go to   Dashboard  Appearance  Templates. 
Create a new template or open an existing template for editing. 
Select any of the checkboxes under Archive Index Templates list to make the template work on archive pages. 

How to output posts listing on the template.

Make sure your template has at least POSTS and TITLE modules on the template. POSTS module will be filled up with the proper posts automatically.