You said “applied for asylum”, so it’s not processed yet. Did you obtain your EAD? Just because they hired you it doesn’t mean you’re good to go. HR will make sure your work permit and any other legal documents are valid and applicable to this job. This is something HR usually processes right before training.
Every airline has different passport policy. No universal rule applies to all. When can you renew your passport?
Side note - I’m a green card holder and it’s also your responsibility to make sure you’re eligible for this job. USCIS will find out and your application will be in jeopardy.
Ask your airline!