السلام عليكم ورحمة الله وبركاته أحسن الله إليكم Am I allowed to teach others Qa'idah Fi As-Sabr once I have completed it under your explanation? Or do I need an ijazat from you?

For teaching your family and close ones that is not a problem as long as u say Allah knows best for what u do not know. As for teaching publicly, it needs alot of years invested in seeling knowledge and reaching a certain level where u can actually teach publicly.