I am no expert in Adobe Illustrator yet there are other forum members that have more experience.
Your set of questions begs another question of what is your final objective for your project.
If a set of AI vector files are already available for use, there is no no need to reverse engineer the soft edges to hard edges. You can start with the AI files directly and get much better results in almost all cases. A better understanding of what you have available are your disposal may help forum members help you with where you want to go.
More specifically to your questions:
In general, AI is a vector based program and PS is a raster based program and those are different beast
That said, AI does support raster images and PS also has vector capabilities.
If you convert your raster to vector (not easy or clean in many cases) or have already used Shapes (which are vector based), those can be exported in vector format and iimported to AI.
Also, vectors in AI can be exported and brought into PS (simplest example is bringing in an AI file as a Smart Object). The sharpness in PS then is a function of the pixel resolution you use and factors such as anti-aliasing.
Note the illustrator still has to be viewed on a monitor (which is raster) or printed on a printer which is also raster. The vector data scales extremely accurately which is one of its main advantages.
I know that it not a crisp answer so the answer is yes with limitation to your questions.
