I finally did it. Some tips on what and how I have done, I hope it helps:

Specifications:
Zen Cart v1.3.7
Google Checkout v1.3RC2
PHP over CGI server (private SSL certificate)

Issues:
1. My orders aren't shown in Zen Cart Admin section/Customers/Orders if the customer checks out with Google Checkout. I've been able to checkout but nothing more. [Done]
2. My shopping cart not clear itself after a live Google transaction. [Done]
3. Products With Attributes Stock v4.4 installed. This is still an issue for me, it doesn't update on itself.

I've had an "HTTP Basic Authentication failed. Can't retrieve Merchant Id/Key, Installed over CGI??" message in response_error.log from googlecheckout directory.

How it worked for me (live mode):
1. CHMOD googlecheckout directory to 777 with Filezilla (right click on the folder, choose "File attributes", write 777 in "Numeric Value" field and pressed OK)
2. Set the .htaccess Basic Authentication Mode to TRUE (in Zen Cart Admin/Modules/Payment/GoogleCheckout) and pressed the Update button from the bottom.
3. Ran the htaccess.php from http://www.yourwebsite.co.uk/admin/htaccess.php
a) Unchecked Sandbox Account
b) Filled Merchant ID and Key with the ones from my real account.
c) Checked "Check if dir exists:" and "Create files:"
d) Clicked "Create"
4. Closed the window and checked on my webserver; .htaccess and .htpasswd files have been created in the googlecheckout directory (sometime it takes a while to refresh, close the connection to the server and open it again)
5. CHMOD googlecheckout through cpanel/File Manager of my server to 755 (this is a must in my opinion in order to work)

If it doesn't work, you must check the response_error.log and response_message.log from googlecheckout directory after you place an order. Also, look for your order number which is listed in your Google account and check in the Settings/Integration/Integration issues what is happening there. On my previous unsuccessful attempts I had 403 and 500 errors and all were due to the fact I didn't really CHMOD the googlecheckout directory properly (from cpanel).

I'm not a specialist and it’s possible to be wrong on some points but it worked for me. So please correct me if this is the case. I've wrote this after I read several pages with no real/direct info on how to get this done and I noticed there are many others in my situation.

*sorry for the way this post is written, english is not my native language