Yes, you may work on V2 visa after obtaining a valid Employment Authorization Document. You must use Form I-765, Application for Employment Authorization, to apply for a work permit.
No, there are no travel restrictions on V-2 visa. As long as you possess a valid V-2 visa you may be inspected and admitted to the U.S. after traveling abroad. If you have a pending Form I-485, Application to Register Permanent Residence or Adjust Status, you do not need to obtain an Advance Parole prior to traveling out of the U.S.
Note: The departure of a V-2 nonimmigrant with a pending application for Adjustment of Status is not considered as the abandonment of the adjustment application upon departure.
The processing time for V-2 visa is anything between three and six months from the date the application is made.