Here's some documentation on hosted content. If you have 10.1 and an ArcGIS.com subscription, you can create a feature service from your shapefile data, and then bring that feature service into ArcMap 10.1 and edit it. This will sync the changes to your layer on ArcGIS.com.
No, there is no automated synchronization. You need to republish the data and recreate the service.
One thing you'll need to think about is where will the "authoritative" data reside? If you're creating a hosted feature service to support editing and also allowing editing against your local database, it will be challenging to synchronize the versions. We don't currently have the set of tools available in the hosted environment. This workflow is currently supported by an onsite ArcGIS server installation synchronizing with the local database. Services are then published out against the same local database.
Thanks,
Mike