Answering either "yes" or "no" directly, is not an accurate answer. The answer of your question can be "yes" if training set and test set are compatible. However, the answer will be "no", if training set and test set are not compatible.
Let me explain in details:
To accomplish your goal, your training set (public dataset) and test sets (your own dataset) should have exactly same dataset features. However, if public dataset has different features from your own dataset, this makes your prediction model not reliable one, and learning algorithm could not even work in some platforms because of this sort of feature differences.
Please let me know if you still have further issues.
yes, this should work, however, it will be good to bootstrap your results to get an estimate of the confidence interval for your predictions on your own dataset, and compare that to the public dataset results, this will give you a hint of the compatibility between your set and the public set.
Such a thing may be possible. An entire research branch, which is called domain adaptation - or transfer learning, is dealing with such a similar scenario.
If this will actually work depends on several factors such as the similarity of the datasets, the transfer learning method etc. The easiest scenario is the one you described, i.e. just using a pre-trained classifier on a new dataset. Nobody can tell if it will work - just try it on your data.
There are most likely already papers out there that do something like that. I would advise browsing the internet with above keywords.