Can this wireless car charger be able to grip thick mobile case while charging?
Can I use this wireless car charger as a phone holder mount?
It's holds the phone if that's what you mean
Can it continue charging the phone when on a bumpy road?
Yes you can and it will be smooth all the time
Can it charge iPhone 11/12/13 with fast charging capabilities?
Yes you can charge iPhone 11/12/13 with this charger.