I hope you are well today and thank you for your question.
Just started testing the lite version of Wiki. The contents of the Wiki need to be behind our membership wall, but currently it looks as though I need to set the permissions on each wiki page by hand to achieve this.
How are you setting the permissions on each wiki page? Are you using membership plugin http://wordpress.org/plugins/membership/
Does the paid version solve this issue? Can I set the entire wiki and all of it's content to require a user to be logged in?
Paid version of wiki plugin will allow you to control wiki editing privileges but to protect the content from non logged in user, you will have to either custom code the functionality or use membership plugin.
Also, on a related topic, my theme uses a multiple page templates. Is there any news regarding the ability to choose which page template Wiki uses?
Currently the plugin doesn't support assigning different custom page templates for different individual wiki pages, you can develop custom functionality to do it like this plugin http://wordpress.org/plugins/custom-post-type-page-template/
You can use standard template pages easily as the wiki uses a custom post type( incsub_wiki ). To create a new template you can follow the WordPress method:
You will need to code the templates up yourself though.