2012 MacBook Air screen immediately falls down, can it be repaired?
Yes - this is eminently repairable and easy to diagnose remotely since you describe a typical and physical hinge clutch failure. 

The good news is you can buy liquid damaged / logic board broken Macs that are vintage for very low cost and possibly know the screen assembly is fine by booting to recovery or just trusting the seller.

Swapping the entire display is pretty easy but to get to it are numerous delicate and fragile connectors likely to break the rest of your mac if you’re not careful or have a helper to hold things. Also, you have to remove the battery, which can be bent / damaged and cause a fire risk.


https://www.ifixit.com/Store/Mac/MacBook-Air-13%22-Mid-2012/Screens


This also is a very easy repair for someone that does repairs professionally and they can source parts cheaper than you in most cases. They are right, you can’t just repair the hinges - you need an exceptionally strong press that costs far more to make than buying hundreds of displays.

You risk breaking the LVDS cable if there is a sharp piece broken, so short term you need to use it in closed clamshell mode with an external display and baby it while you decide if it’s worth $300 to fix.