at 10 days, it might be done fermenting, but it most likely needs some time for the yeast to finish cleaning up after themselves, unless you pitched a large starter or brewed a really low gravity beer... Give it another week or so at least, then go ahead and keg it up. To answer your question, no, it won't keep conditioning, much, once kegged and in the fridge. Fridge temps tend to cause any remaining yeast to drop out of solution and go dormant pretty quickly, so they won't be doing anything for you.
And shaking the keg actually does work to dissolve co2 faster - i don't quite understand the physics or chemistry involved, but it works.
