Varnish Headers

Max-age is a common browser caching mechanism. When a server sends it, the browser will not request the URL again for 600 seconds. When a browser sends max-age=0, it wants to revalidate its cache entry (usually because the 600 seconds are over, or the refresh button is clicked). This is useful for static content that isn't updated such as images.

An expire in the past is a useful way of not allowing the browser to cache the request. Note that just because the browser is not caching request, doesn't mean the page is cached on the server. In fact, this header is required for dynamic content to detect if the page has been updated.

How often is Drupal cron running? This clears the page cache.

Your headers all seem correct; just you have a short ttl (10 minutes). This short time frame is needed because your site might change and it needs to be updated then. Using the varnish module allows you to have a longer ttl. I do have the Cache Expiration module that will allow you to try and have even longer page lifetimes although its not ideal ATM. If your using boost I keep everything in the database and that allow me to keep an accurate record of what content shows up where (if your using the latest dev). Eventually I will port over all the changes from boost into the expires module; only issue with expires is it will then require 2 database tables to keep track of everything (just like I do for boost). I've been trying to avoid database tables but it appears that I need them if I want to do advanced cache expiration; like I do in boost right now.

Examples of advanced cache expiration include (things that require a database):
flushing the view on node edit/create
flushing the panel on node edit (nodes loaded up in via context)

You can also try setting the cache for the page to a longer ttl (1 day), then use Edge Side Includes (ESI) to insert into the page elements that refresh more often (such as blocks or the main content area in the home page).