It appears from the video the sink bottom connection is assembled wrong. From top to bottom you should have sink, rubber gasket, metal flat washer, brass nut. This or the P-trap connection do not need to be super tight, firmly snug should be enough. If the connections leak when snug, the gasket or sealing ring is likely damaged, or possibly their mating surfaces are not clean and smooth.
More disturbing is you say drain/popup detached from the sink when you were fussing with the P-trap? This should not be possible. If it did happen, the drain/popup is broken and needs to be replaced. This is not something you could have done alone, it was ready to break anyway. The flange you see when standing over the sink is part of this assembly, the large nut underneath sort of clamps the sink between the rubber gasket and this flange. The drain/popup is normally installed by dropping it in from above. If it came out from the bottom it is broken.
Sometimes these charity facilities have work "donated" to them, and it might not be the most experienced plumber, electrician, or whatever doing the job. You might want to get a trusted, experienced plumber to look over the install to make sure there's not something fundamentally wrong.
That said, and assuming the vents are installed correctly, the design of your particular vent will determine if it can be cleaned or not. In a lot of cases a plumber would just replace the vent because it's not worth their time to try and clean it, and hope it still works. If you're a full time maintenance person at the facility (or a DIY'er), you have time to try and clean them and then replace if they don't work since you're there all the time any way (no trip charges, driving, etc).
For instance, in this article, the guy just replaces the vent because it was so nasty, but maybe he could have soaked it for an hour and scrubbed it out. Or maybe he just knew it would never work right because he'd done it before...
Short version: Try to clean them if you think it's worth it, but you might just have to replace it anyway.
Best Answer
Cut the extension pipe off close with a hacksaw.
Then use a regular socket on the brass nut.