Yes, most of the large models that are not feasible to be trained by individuals are downloadable. Frameworks make it easy to import and export snapshots of model weights. See https://github.com/tensorflow/models/tree/master/research/sl...
Yes you can. The "universal" format is ONNX. You can even do crazy things like this: https://blog.owulveryck.info/2018/06/11/recurrent-neural-net...
For example, if I want to recognize pictures of animals, with reasonable accuracy, how much space would such neural net take?
I'm not very taught about ML, but in my mind, since ML is generally heavy on computation, it should be possible to distribute neural nets for specific applications and re-use them.