The 404 is usually resolved by setting your Permalinks back to the default setting, then immediately changing them back to what you have them set to. This causes the permalinks to be rewritten and usually takes care of it.
The timeout after 5000 milliseconds means that your WordPress site has requested the HTML for your form and the expected response has not been received before the timeout is reached. 5000 milliseconds is a very long timeout so this means your WordPress server and Google aren’t communicating. This problem can happen if your Google Form isn’t published (shared) so the URL of the form can be resolved without having to login to Google.
No idea what the 508 error code is, I’ve never had anyone report that one before. A quick Google shows this error is fairly common on low end hosting plans. Does your hosting provider limit access to certain resources?
The Google Forms plugin is based on the WordPress HTTP API. All remote requests to and from Google are made through the same API. The timeout is defined by the API bit can be increased on the settings page in the plugin.
As for detailed documentation, like many plugins this is something I do in my spare time. I do my best to answer questions quickly and completely but due to work and other family commitments, sometimes it takes me longer than others.
There is a lot of stuff on my web site and there is a lot of stuff here in the Support Forum. Is it complete and well organized? Nope. That said, getting the plugin up and running using a simple form should be as simple as what I outlined above in most cases.