Will AppStore applications continue to work after an OS downgrade? Our daughter has an old MacBook Air (2010 model) that worked fine until a few recent OS upgrades. Now it's almost unusable with its 2GB RAM that can't be upgraded, so she is considering cleaning her MBA and reinstalling an old OS.

I wonder whether this is going to work smoothly in relation to apps she was using. Will the apps she will install from AppStore require her to upgrade the operating system?
Minimal OS version requirements vary from application to application, so there is no general yes or no. The minimal OS version is part of the application description in the store though, so the best approach here would be for your daugther to look at her key applications in the store and check whether all of them will work with earlier OS versions as well.