long-lived access token

These are the steps that were previously in the question - they have been migrated to this answer.
Having found that it is possible to generate a Facebook Page Access Token that does not expire (with help from @Igy), here is a clear, step-by-step quide for all those looking to the same:

Make sure you are the admin of the FB page you wish to pull info from
Create a FB App (should be with the same user account that is the page admin)
Head over to the Facebook Graph API Explorer
On the top right, select the FB App you created from the "Application" drop down list
Click "Get Access Token"
Make sure you add the manage_pages permission
Convert this short-lived access token into a long-lived one by making this Graph API call: https://graph.facebook.com/oauth/access_token?client_id=<your FB App ID >&client_secret=<your FB App secret>&grant_type=fb_exchange_token&fb_exchange_token=<your short-lived access token>
Grab the new long-lived access token returned back
Make a Graph API call to see your accounts using the new long-lived access token: https://graph.facebook.com/me/accounts?access_token=<your long-lived access token>
Grab the access_token for the page you'll be pulling info from
Lint the token to see that it is set to Expires: Never!
That should do it. You should now have a Facebook Page Access Token that doesn't expire, unless:

You change your Facebook account password
You lose admin access for the target page
You delete or de-authorize your Facebook App
Any of these will cause the access token to become invalid.

If you are getting (#100) Tried accessing nonexisting field (accounts) on node type (Page), go to the Access Token Debugger, copy the value of User ID, and use it to replace the "me" part of the URL in step 9.