A person who is in proceedings and appears to qualify for V status can request the Immigration Judge to administratively close the proceedings, and be given the permission to apply for V status. On applying, if the USCIS finds that the person does not qualify for V status, then the proceedings may be recommenced.
If the foreign national already has a final order of removal, deportation or exclusion, the person must follow the usual rules with regard to whether the proceedings can be reopened. The regulations include deadlines for motions to reopen, and there is no special provision for a person applying for V status. One exception to the motion deadline is if the USCIS agrees to join the motion.
Yes, you may apply for Adjustment of Status to permanent residence while on V-1 visa if you meet the requirements of INA section 245(a).
Another option for V nonimmigrants who do not meet the requirements of INA section 245(a) is to adjust under the temporarily restored section 245(i). Section 245(i) allows many immigrants with priority dates earlier than April 30, 2001 to adjust in the U.S..