Discuss if other family and friends are to be invited then provide a list to the groom's family so they can send the invitations. (Do check with your daughter and her fiance first, before making a list, to determine just how many and who is to be invited as you don't want to put a strain on the groom's family who may be working with a budget for the rehearsal dinner.)

It's always a good idea to send invitations, if for no other reason than the wedding participants will know there is a dinner and where it will be. You could even put on the invitations that there will be no children, and what ever you feel you need to put so that there will be no problems with the rehearsal dinner that night. This way you are letting them know ahead of time, so that they can make arrangements for the night of the dinner. But since the groom's family is paying for the rehearsal dinner, you have no say what so ever when it comes to who is being invited to the dinner. You are just going to have to deal with it no matter how bad it may seem to you.

The day of the wedding may be different and since both sides of the family will be paying for the wedding you can sit down with you're soon to be in-laws and let them know how you feel about something's. Tell them about some of the things that you don't want to happen the night or day of the wedding, just so your in-laws know how you feel and you can work out how you plan to write it on the invitations.

As hard as it may be, it seems like "family" never seem to understand that "no children" includes them.
You are dealing with a doting grandmother, who obviously thinks there's no one who takes presidency over the children.

Since the rehearsal dinner is hosted and paid for by the groom's parents, I think you are going to have to bite the bullet on this one. It's called compromise. If the children come to the rehearsal then they probably won't be at the wedding.

You might put a bug in the groom's ear about possibly having a sitter at the dinner to entertain the kids during the toasts and the better part of the evening. I'm surprised that his brothers and sister-in-laws don't understand they are part of the wedding party and need to focus on the bride and groom and it's an opportunity for them to enjoy the day without the worry of the kids.

I'd say, let this one pass. Unfortunately, its things like this that set the mood for the relationships of the families over the next few years. You need to be supportive of your daughter and enjoy the day as well.(Hopefully she has a wedding coordinator to handle the day so you don't have to.) On the off chance the in-laws may bring their children to the wedding, you may want to go ahead and secure the services of a babysitter.