Is allowing colleagues to drive the company car for work-related tasks during official hours, and our use of the car after hours for daily necessities, considered a betrayal of trust, and will our prayers be accepted when going to the mosque with this car?

If your manager—who is authorized to do so—permits you to use the company car for your personal needs after hours, or allows other employees to drive it, then there is no harm in it for you. The key factor is the permission of the responsible authority. The use of the car does not affect the validity of your prayer.