When can I enter an employment without needing the approval of the “Bundesagentur für Arbeit” anymore?
That depends on where you are living in Germany. In Berlin there is no need anymore that the Ausländerbehörde asks the Bundesagentur für Arbeit. Before they had to make sure that there is no German or EU-citizen that could do the job as well (which is still the case in other federal states). Now they only check the content of your work contract with a focus on the working conditions. So here in Berlin the procedure is quicker than elsewhere.
Keep in mind that it is always the Ausländerbehörde that decides if you are allowed to work or not. Except for those people that have the comment “jede Beschäftigung erlaubt” in their papers.
In other cases, for example when you have an Aufenthaltsgestattung, it might say “Praktika gemäß Mindestlohn gestattet”. Than you can do (paid) internships. Or it says “Arbeit nach Erlaubnis der Ausländerbehörde”. So you are allowed to work, but you have to go to the Ausländerbehörde with your contract first.
Apart from that you can for example do certain forms internships that do not require the agreement of the Bundesagentur für Arbeit or certain other forms of preparation and specific social volunteer jobs.
The Bundesagentur für Arbeit is currently planning to create 100.000 so called “80-cent-jobs” that people that are seeking refuge can do while still being in the process. These jobs are quite similar to the 1-euro-jobs that are already existing for permanent unemployed people. They are quite low paid, but it is better than nothing. The program is probably going to start in October.