Nice to hear from you again!
As you most likely know from first-hand experience, working for a Zeitarbeitsfirma means that you only work on temporary basis. Your workplace and tasks can change quite often which can be stressful and you might earn less compared to the regular employees (of course there can be exceptional cases as well).
In general I would say that it is a good idea to switch to Festanstellung if you are happy with your job, the company, the payment and the general situation because it makes it easier to plan ahead. Carefully check the contract that they are offering you to see if it improves your current situation compared to being employed at the Zeitarbeitsfirma.
Keep in mind that you have a valid contract with your Zeitarbeitsfirma. If you want to work at your current workplace permanently, you have to come to an agreement with your Zeitarbeitsfirma because they will lose you as an employee.
Concerning your last question: since your case hasn't been decided yet I assume that that you asked your competent Ausländerbehörde for a work permission, right? If you switch from one type of employment to another you might have to ask your Ausländerbehörde for permission for this new contract again (even though it's the same tasks as before). But I'm not sure about this. Maybe @Marcel can help us out here. He is a lawyer and knows a lot about the legal framework of labour market access.
I hope this answers your questions. Feel free to ask if anything remained unclear!