Hi @Hellothere
The ‚Erklärung zum Beschäftigungsverhältnis‘ is enough for the work permit.
Each Ausländerbehörde works a bit differently. In some places, you can go in person and submit such an application. Others ask you to send it to them. I cannot say for sure what the usual procedure is at your local Ausländerbehörde but generally per email is fine.
Yes, you will receive a written decision called a ‘Bescheid’ on your application to work. They will send this to you by post.
Your employer will take care of your Sozialversicherungsnummer. You will have to tell your employer which health insurance company (Krankenkasse) you would like to join and you will be registered there. The Krankenkasse will then organise you to be issued with a Sozialversicherungsnummer.
The tax number gets sent to you automatically after you are registered living where you are. If you have not received it or no longer have it, you can ask for tax authority to send it to you by using this online form:
https://***.bzst.de/SiteGlobals/Kontaktformulare/DE/Steuerliche_IDNr/Mitteilung_IdNr/mitteilung_IdNr_node.html
You can start working already even without the tax number. For up to three months, your employer can do a preliminary tax deduction on your wages. If your employer has doubts about this or requires more information, you can show them this document about tax number registration from the tax authority:
https://***.bzst.de/SharedDocs/Downloads/DE/IdNr/hinweis_idnr_gefluechtete_deutsch.pdf?__blob=publicationFile&v=3
Best,
Éanna