Can generic data sets be suitable for specific sentiment analysis
It depends.

You're basically asking if your sample (training data) is representative of the population (all written words). 


Are you doing sentiment analysis on movie reviews? It'll work great.
Are you doing sentiment analysis on TV reviews? It'll probably work great.
Are you doing sentiment analysis on book reviews? I would give better than 50-50 odds it'll work.
Are you doing sentiment analysis on Twitter posts? Now we're getting shaky. People tend to write much less, use less formal language, and use more emojis which your movie review model wouldn't have seen.


That being said, there are definitely "generic" sentiment analysis services like here. Try out your model against Algorithmia on what you would consider a generic set of data (e.g. a bunch of tweets) and see how it does.