Free and paid versions are completely different and they use different architecture and approach. I could answer to your question, but it is very technical and I'm sure that you are not interested to go into technical details.
Here you will find how to set a cookie domain:
http://docs.joomla.org/J3.2:Global_configuration it is very easy to do.
I understand that you want it to happen and I thought that you would just search how to change cookie domain and find the solution yourself, so I didn't go into the details.
I understand that you are not a webmaster and shouldn't know all the technical aspects, but if you have any questions I'm always ready to answer them.
The free version works per request and does the translation then forgets until there is another request. While Pro and Enterprise save the translations and make it possible for you to make modifications. We need to discover your content to be able to translate. You can speed up the discovery process by manually browsing your website in different languages, so everything will be discovered and get translated. If you even don't do that, it is not an issue, crawlers who crawl your website will do the job for you.
The GTranslate Free version you are using is a language switcher only in case you have configured it to work with Pro or Enterprise version. However if you manually enter in the address bar es.sim-x.net your site will get translated by Enterprise version.
Please let me know what you want to achieve and I'll try to make it happen for you. As I understand you want your website to get translated and indexed in search engines. If that is the case, then it is set up and ready to use (except the SSL which you need to send, so I can setup for you), you can turn on Enterprise option in free module and make manual corrections to your translations.
Let me know.
Thank you!