Are there any cases where a person has not been deported any various reasons? What are the common ones.
Also in a process of removal, is there a way to gain any status to remain in USA?
Also what are the reasons for out of status people who get deported? Is it the Criminal charges?
Yes ther are cases in which people apply for relief from removal ( Asylum, cancellation of removal). although it is somewhat harder to obtain or adjust status through this process but not all the times.
I know people who went through the process and gott GC.
you are required to ask for relief for which you think you are eligilbe or you can make the case presentable to immigration judge. you will be given enough time to come up with documnetation regarding your claim and if you are good at articulating it, you will sure win the case.
Most commonly deportation results from felonies or national security related grounds, however moving violations sometime does corroborate other evidences.
In order to determine what you can do, I need to know wher exactly you are standing in immigration process. From there, we can strat building the road.
Good Luck.