What Moderator (default role) permissions allow a member of this role to create a new blog post?

You need to consult the blog itself to see what roles have the create/edit permission

Hi Patrick.  Can you tell me which permission above would allow for creating blog content?  None of them say anything about 'create' or 'edit' blog or other content.

Also, do you know whether any of the permissions above are not set/checked by default or OOTB?  When I impersonate the user and remove him from the Moderator role he no longer has access nor can see the 'New' button to create a blog post.

Hi Patrick.  Can you tell me which permission above would allow for creating blog content?  None of them say anything about 'create' or 'edit' blog or other content.

Also, do you know whether any of the permissions above are not set/checked by default or OOTB?  When I impersonate the user and remove him from the Moderator role he no longer has access nor can see the 'New' button to create a blog post.

if you are expecting to see blog permissions at the site level in administration, you won't.  This is only for site level operations.  Permissions for groups and applications are managed through the permissions area of the managment panle on each group and/or application.  Meaning the blog in question has to be accessed, enter the management panel via the pencil, and check the permissions.   You will see blog permissions listed for group roles and site roles(potentially).  Same applies for the group the blog is in, you will need to review both for the create blog posts/edit blog posts permission.

I would suggest reviewing the documentation
How can I control which actions members can perform on a blog?
How do I change permissions for members of a group?https://community.telligent.com/community/12/a/user-documentation/c/groups/UD210/default-site-group-and-application-permissions-for-roles 

Thanks Patrick M.  I found that the user was able to post because of having the Blog Moderator role assigned and the role was allowed to create posts in the group/blog.  We removed the role from the account so it is now unable to post there.