There are many startups which do this (or at least similar) already. It's often referred to as "backend as a service" (BaaS). Some links below, but you'll find many more via Google.
Yes. Firebase. I think Firebase is interesting because they focus on realtime data which is crucial for a lot of apps these days.
Strongloop has http://loopback.io which has been interesting to work with. Looking to free up some time to investigate a little further!
there's a lot of energy/activity with the strongloop project - I am looking at that as well... and now Cognito will demand my attention, too :)
Orando Labs (https://orandolabs.com) has EnduroSync and Identio. Striking similarities between the products. EnduroSync is a full object store and does not have the data size limitations.