Allow public to edit content type
I'm not sure I understand you correctly but let me clarify:
A.If you have a new content type that users can vote for that. You just use the votingapi http://drupal.org/project/votingapi and use with Fivestar module http://drupal.org/project/fivestar or plus1 module http://drupal.org/project/plus1. You will go to plus1 or fivestar for setting the content type that be voted.
B.To determine the total of the votes of your content type you should do following:


Create a new view and choose the type of your content, remove sort options, fields and other default settings.  
Add a field for "Content: Type" Expand the "advanced" part on the right and set "Use aggregation" to "Yes"  
Create the relationships with votes table("Content: Votes")  
Add another field for "Content: Votes", click "Aggregation settings" and set the aggregation type to "count" 
The second "Content: Votes" should now look like " COUNT(Votes: Value)"  and now you can see your content type and the total votes of that.


C. In order to anonymous users can edit your content type just go to permissions and set configuration for anonymous users.