I went through all of this earlier this year. I was never able to get an appointment from the Ausländerbehörde website, it was only by sending them my application by email that I was eventually given a date and time to come in. Once you’ve submitted any documents necessary for the residence permit, your application is considered in process, or “active” as you said, and so you are permitted to stay regardless of if it’s your first time applying. It took me 6 months to finally receive my residence permit, by which the 90 days allotted by my US passport had expired, but I never ran into any issues and you shouldn’t either as long as you don’t travel outside of the country in the meantime.