Depends on the terms of your custody agreement. Unless you were fools you surely made some arrangement for who has the kid when the custodial parent is unable.

No. No parent has the authority to "give" custody to someone else. Custody is a LEGAL status.

Yes, she CAN leave your son in the care of her mother while she is away. This is her responsibility as the CUSTODIAL parent. To ensure the child is cared for by a responsible adult in her absence. YOUR parenting time will continue to take place on weekends, as outlined in your court order. During YOUR time, YOU may leave your child with any responsible caregiver in your absence.

If your ex needs a week away so she can regroup, then I would say it would be beneficial for all. If she is letting a retired grandparent care for him, then I bet your son will be having a blast. Don't worry about it, you will see him on your weekend as usual.

If you're remarried and your current wife doesn't like it, then explain to her that everything will be Okay, and life will continue as normal.

If you want your son to stay with you, why can't you ask your ex about it?

Without reading the custody/visitation agreement there is no way to know.

Anything is a guess.

The only reason she would NOT be able to do this is if there's specific wording in the custody agreement that your son is not be left with his grandma.   Odds of this are very low.

Maybe it's time to challenge the existing custody agreement.  If you want your son and he wants to be with you, she's using her son as a weapon to get back at you.   Courts don't look kindly on this stuff.  Maybe you should be the custodial parent.

(Obviously, this assumes you're a good parent).

Read the custody agreement. It's all in there.

I wouldn't think so unless a judge awarded it to her. If it is just one week abroad I wouldn't try to make waves and let it be and see your kids after she comes back. The kids might benefit from spending time with grandma.