I'm using QGIS 2.4 and got some issues with the print composer. First thing is that the "update preview" button reloads the map, but does not change the view when I move to another place in the main window.
First, in QGIS composer map items have an independent scale and extent to the main canvas. Panning or changing the view in the main map canvas won't alter how the map appears in the composer. To resync your composer map to the main canvas:
- Select the map in the composer
- In the map item's properties panel, click the 'Set to map canvas extent' button under the 'Extents' section
In regards to your offset print issue - the OpenLayers plugin is a third party plugin (not part of QGIS itself) and has not been written to work with the map composer, and experiencing offsets between the Google maps layer and other map layers is a known issue.