I've registered a folder that contains a .gdb with some feature classes with ArcGIS Server 10.2 and exposed the data (read privileges) to the ArcGIS Server domain user account. I'm using these feature classes in a published map service. If I alter an attribute or the geometry of one of the feature classes (in arcmap on my local) and save the changes, will those changes go live in REST Services?
It definitely should. One of the ways to test it is to publish a map service and access it with REST Service Directory JS API Viewer. Then go ahead and edit the attributes and geometry of a certain feature.
If this attribute is used as the symbology field, after refreshing the page or zooming in and out (forcing the map image to re-draw), you will see feature with another symbols ('cause the attributes were changed). You should also see the geometry updates.
One thing that quite many users may notice is that sometimes after editing the geodb data in ArcMap, the map doesn't get refreshed when viewing the map service. This is because you see a static image that was generated with the data as they were at the moment of request. To update the image, you have to send a new request to the server which can be done by zoom-in and zoom-out or panning between the areas outside of your current visible extent.
If doesn't happen, try refreshing the whole web page. If this doesn't happen, clean your REST cache (even though it is supposed to be cleaned automatically). But at this moment you should start troubleshooting the map service...