If you happen to get the access denied for anonymous users on a webpart (in my case list view) and you already gave anonymous permission on the list, you are not alone.
Apparently this is happening especially when you use custom xsl for the display of the items. Luckily I came across this thread that shows the trick to avoid this bug.
You just need another xsl that imports your custom xsl. In the webpart Xsl link you will then use this other xsl instead.
There you go. I tried this and I'm not getting the access denied anymore.
Home » SharePoint 2013 » Sharepoint 2013 List View Webpart Anonymous Access denied error You do not have permission to perform this action or access this resource