You can apply after you have been a Lawful Permanent Resident for 2 years and 9 months IF you got your Green Card through marriage to a US citizen and are still married and living with the same citizen spouse. Otherwise, you apply at 4 years and 9 months.
First, so long as he appears at ALL immigration court hearings he will not be arrested simply for not having a lawyer. You need to contact his criminal defense attorney regarding the criminal case - it could be the charges were dropped or it could be the case is ongoing. Also consult with an immigration attorney who is familiar with criminal consequences on immigration.
It appears that you "waived" the clock the day you submitted the asylum application. That is why the clock shows as "0". Unfortunately, you need for 150 days to elapse before filing for work authorization. Unless you an get the clock restarted, you will have to wait until your application for asylum is approved before you will be given work authorization.