This is a very personal decision, and there are no right nor wrong answers. You are welcome to stay for as much of the process as you feel comfortable. There is no judgement if you prefer to say goodbye and step away prior to process.
You can absolutely hold your pet, either on the floor or on your favorite sofa or comfie chair. We do need to be able to get near your pet, so recliners should have foot rests retracted. Coffee tables should be moved to allow us to be on the floor near pet if pet is on sofa.