As long as the project with those issues support anonymous access, you can do that. It will open up all issues in the project with anonymous access unless you implement issue security scheme.
Following links might be helpful.
Jobin has already answered perfectly on your first question.
For your second question: yes it's possible to publish that list on your Confluence server.
Simply use the Jira Issue Macro, insert the JQL you use for your anonymous search (or copy and past the url you get when you export the searchresults to XML). Make sure that your confluence page can also be viewed anonymously.
Some extra info can be found here :
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