Thank you for using the platform so actively!
With your national ban on deportation status, you receive a residence permit for one year (that can be extended). That makes it possible for you to rent your own apartment. The job center or Social Welfare Offices will deem what type of housing is "appropriate" for a couple with two children.
Depending on that decision, you are also going to receive financial support, either from the job center or the social welfare offices of your area. Please note that, if you receive social benefits, for the first three years you may have to reside in the area where your asylum procedure took place, due to a so-called "Wohnsitzauflage".
Have a look at this Berlin Service website (just bear in mind that some things are specific for Berlin) for more details on social benefits: https://www.berlin.de/fluechtlinge/en/information-for-refugees/benefits/
Here are some BAMF sources on How to find housing, information on rent and moving in: http://www.bamf.de/EN/Willkommen/Wohnen/wohnen-node.html
Also have a look at these threads from our platform for some tips - there are even more threads like this if you simply use our search bar on the right side!:
how can I find an apartment in Berlin?
What documents do I need to apply for an apartment?
Please contact the job center/ social welfare office of your area for further information on necessary forms and social benefits for your family. You can also reach out to a refugee support service, who usually have people who can help with finding housing, going to viewings and with the documents.
I hope this helps. Feel free to reach out to us again if you need any further assistance :-)
All the best,