Should i take a video of my defective touchpad now...
Yes — take that video today, and don't wait until after exams to open the case.
Show the shipping label or box with date, and the Surface's serial number on screen or via Settings > System > About.
Do side-by-side clicks (left corner vs right corner) with the mic close enough to catch the sound difference.
Mention the charge-cycle count on screen (from the battery report) as supporting context that it's refurbished.
Either way, it's a hardware/build-quality issue, not user damage, and it's exactly the kind of thing warranty replacements are supposed to catch in QC and often don't.