This depends upon the law in your state. In my state it is allowed unless the divorce decree prohibits it (rare). Otherwise, a noncustodial parent would be stuck vacationing in the same state for many years. I would notify the other parent in writing as early as possible prior to the vacation, if possible, and keep a copy of the email or letter you send.
Custodial parental permission.
Yes, as is their custodial right.
Unless there are court's restrictions, yes.
It is against the law for a non custodial parent to take their child out of State (can be considered kidnapping) without the written permission of the custodial parent. If the custodial parent agrees then be sure it is in writing, with the date and signature of the custodial parent and carry it with you during the trip.
If a custodial parent can take a child out of the country on vacation without telling the non-custodial parent will depend on the court order and the laws in the state they live in. Many states will want the custodial parent to get a notarized document from the non-custodial parent stating they are giving their permission for the child to leave the country.
I would be necessary to give the other parent notice of the pending vacation plans. Whether or not you would need permission from that parent to take an out of state trip depends upon the terms of the custodial order and the laws of the state in which the order was issued.
Sole physical AND sole legal custody? If so, most likely as long as the visitation does not interfere with the non-custodial parent's court ordered visitation. If it does, then the parent planning such a vacation must secure permission from the other parent, or permission from the court beforehand.
It depends on the court order you have with the divorce. The non-custodial parent should have a letter from you giving permission to take the child across borders, including internationally.
In Michigan, the custodial parent can't move the kids more than 100 miles away from the other parent and can't go out of state, even for a vacation, without permission from the court
The non-custodial parent must notify the custodial parent of their intentions and the destination of their vacation. The custodial parent has the right to know where the child will be. Also, it is unlikely the visitation order provides that the non-custodial parent can keep the child out of school for a vacation. They would have no automatic right to do so. They would need the consent and cooperation of the custodial parent.
Signature on the Visa Application