I would advice, enter for the 2 weeks using the Visa Exempt Entry, talk to the schools and Immigration to see what is needed to get you the visa by conversion of Visa Exempt Entry if possible,
Then visit Germany, come back with new Visa Exempt Entry and do the conversions
OR
If not possible to conversions, entry for the 2 weeks using the Visa Exempt Entry, talk to the school, get all documents needed and apply in Germany for the correct visa's for all of you.
I would advice to do at your local Immigration Office as the re-entry permit is valid as long as your current permitted to stay, no rush and no need to be nervous.
You mentioned: " ( if I have all necessary stuff ready , money in my bank acc , papers etc ? ) ". You can apply but you must meet all the requirements and the minimum of 2 months money in the bank.
What you show has nothing to do with a 30 day Extension of Stay, which is a big stamp in your passport with dates and signatures.
This is a receipt of notification of your address.
Search your passport for the Extension of Stay permitted up to xx/xx/xx or go back to Immigration and apply again. Do not leave before you understand all.