V visa no longer exists. There is no K visa option for green card holders. Sorry.
General Green Card
When the I-829 fees receipt is issued, it should state that your green card is extended for one year and travel and work during this time is permitted.
You lose your green card status. Reapplying or trying to get a returning resident permit would be the only two options that I can think of.
Basically two ways: through a job offer in your field OR through your own qualifications, but only if you are nationally or internationally famous.
Discussing allegations of fraud and consequences when a short term visa holder tries to convert to green card or another long term visa from within the US.
Three options: Extraordinary Ability Aliens, National Interest Waiver and Investment (EB-5).
What is an EAD?
Certain aliens who are temporarily in the United States may file a Form I-765, Application for Employment Authorization, to request an Employment Authorization Document (EAD), which authorizes them to work legally in the U.S. during the time the EAD is valid.
Why does my new EAD look different than my prior one?