I would like to display my photo albums and photos that are in FB on my own website. However everything I have tried requires an access token that I seem to only get if I lo
Facebook has recently launched a way to share the album with public without having to Signin. Every album will have link below this key word Share this album with anyone by sending them this public link. Take a look at this HOW TO for more information.
EDIT: However, for embedding refer this SO which explains how to grant permanent access to your photo albums
Another option is using 8 photo restricted Facebook Photo Badge
Showzey seems to be working on liberating your photos and provides widgets.
May be this method could help you guys.
You can combine galleria (open source javascript image gallery framework) with a plug-in to integrate with facebook.
You can find the links bellow:
http://galleria.io/
http://www.alexanderinteractive.com/blog/2012/03/display-facebook-photos-on-your-website-with-galleria/
I'm using this very nice plugin for Joomla. Take a look.
Facebook has a badges page that has different options for embedding Facebook content onto your site. The photo badge should fit your criteria.
You can do it through a public fan page or a business page.
I put this script together a while back: http://roughgiraffed.com/FBalbum/. Working on a more legit version - will post when complete.