possibly, ultimately no one knows what the immigration officer in question will think. I'd be more wary about your university deregistering you from their course and then charging you for the course again.
It's possible but it's also very much driven by any potential future employer and the local immigration office. Some can/will do it whilst others will tell you to go overseas and apply.
You will most likely have to use the 30 day extension before being given the 60 day unable to travel due to Covid extension. No harm in asking for it still.
I believe you can go 15 days before your current stamp expires but I'm sure someone will correct me if I'm wrong there. Again, if wrong someone will correct me but the 30 days are added onto the current expiry date.
Not sure which immigration office you go to, there are two. The search function will help with this as it's asked fairly often.