can not use ToolingAPI to save/modify APEX
MUST run all test EVERYTIME a single APEX is saved
False. For a few releases now, you've been able to specify RunSpecifiedTests during deployment; as long as all classes tested cover at least 75% each and all triggers at least 1% each, deployment can succeed. This allows you to deploy a single class that has a single unit test without running all tests (RAT).
Is the Trial-phase the reason why I actually can do both things on this org?
Yes, it's the same as a Sandbox or Developer Edition org.
if the Trial-phase is the reason: is there a way to get the relaxed way of trial-phase back somehow after the regular contract of our client is active and the trial-bar in the top is disappeard?
No, you have to go through the standard deployment process. Why? Because it protects your business from accidental changes that render your org unusable. Some of salesforce.com's largest clients do millions of dollars in business every day on Salesforce, and losing your org for even a day could result in catastrophic losses. I don't have an exact number for our org, but a back-of-an-envelope calculation suggests a bad deployment could easily cost us $10,000,000+ a day in revenue. Trial mode also has a number of other subtle differences, such as limited emails-per-day, that would make it unsuitable for running a business.
is the difference of this behavior documented in detail somewhere?
Not specifically, though a few articles, such as the one in the prior paragraph, do go in to limits regarding a feature when in trial mode as opposed to full production mode.