Some Basics

I had few emails asking some basic questions, and to make usefulness to everyone I'm posting my replies here along with the latest versions.

Well first of all, The HOST STATIC IP in the server configuration should not be "localhost".
Your magento installation is at your remote server(most probably) while your checkout installation is on your computer, what you're telling magento is to try to connect to the postgresql installed on the same machine as magento which is incorrect.

I highly suggest you try everything on your local machine.
A few tips on getting this host locally:

Make a backup of your magento store from System->Tools->Backups.
download it from your var/backups on your magento server.
Then compress your installation files to a zip file (you can skip the media folder coz it must be big and full of images), you can skip var/backups because it will be full of your database backups, and you can skip the downloader folder too.
Then host this magento installation on your computer (use MAMP, it's great).
use "Sequel Pro" to connect to magento's mysql database and make sure that you change core_config_data table the "base url" and "base secure url" to "localhost". if you don't do this, magento will not display properly on your browser because it will not be able to find its required files (css and images)

This way you can test further and be sure you will not break your real shop.

One thing to clear out. At the end, when you want to make the true connection with your live installation and your checkout installation, you must make sure the machine where checkout is installed does have a STATIC IP. This is a must so that your remote server will be able to find it and connect to it each time it needs. Or else you will not be able to work live.
Probably if you do not have STATIC IP, your ISP will charge your for it a little but not much.

The postgresql database you should be connecting to should be Checkout App's. And if everything's set to default when you first created Checkout App's DB then the port should be 5505. Also if Checkout is installed on your computer and your magento is installed on the same one then the host should be "localhost" or "127.0.0.1" depending on if connecting through socket or not.

Going to Administrate stores was meant to show you Checkout App's DB name so you put it in the configuration of the extension, but I was not in focus that time that I wrote the code to use the DB name set in the configuration...so to bypass this issue, put in the configuration DB name: postgres.

After you get the Administrate stores connecting to your server and retrieving your Checkout DB name, put it back in the config.
Administrate stores will also help you set your store order number.

If you go through this and you face other issues, send me back, i'll do my best to help you with your project.

Version 1.3.2Beta
http://www.megafileupload.com/en/file/240535/CheckoutAppSync-1-3-2-tgz.html