If a loan company never has the application for notice of lien signed by the auto owner can they take your auto away?

Sure ... as it is not YOUR car ... you are not the OWNER of the car ... the loan company OWNS the car outright. If you do not pay for something that you buy on time, the loan company has every right to take it back. You see, they hold the owners title on the car, so it is theirs to take away if one does not make the payments anymore.