"Can we download this "Antrag auf Arbeitsgenehmigung" form from Internet ??", no because it's more likely that they will be giving you other form with that, it depends on the city. Köln e.g. gives 2 more forms to be filled by your employer, but I've never had that case elsewhere, like Berlin or Frankfurt.

For the 2 other points, you have to refer to the Ausländersbehörde and ask them about the isssue, their answer on this is more appropriate and if there's something cushy, ask them to confirm what they will say on paper.

D.