Moving a Drywall & Trim Access Hatch?
Yes, you can do as you describe. There's no need to double the cross blocks, as the span is short and the only real weight on the truncated joist is the drywall, which partially carries itself across the gap. Just be sure to nail the cut joists into the cross blocks well. 

Yes, normally attic accesses are built parallel to the trusses. There's no cutting then, just the installation of two cross blocks.