I have my German residence permit renewal interview soon. But since I am sick in my home country, can I reschedule it to be here? Is it possible?
As pointed out by gala, residence permits are not processed within, or are interviews conducted at, embassies or consulates abroad. Managed by Das Bundesamt und seine Aufgaben, interviews are conducted by Ausländerbehörde, according to locale.
Asking for a residence renewal when you are not in Germany, but in your home country on medical leave and receiving treatment (per your separate question), makes it unlikely that the an extension/renewal would be considered. With an illness that has postponed your return to Germany (and may necessitate a new medical exam and certificate), you should contact your school to confirm that you are still enrolled, that your visa status hasn’t been affected by your absence, and that you’re are still eligible to renew residence.