Im going to assume since you said you are a "mom with a camera" instead of a professional photographer that you have little experience in these type of situations. The lighting will depend on the auditorium, the ambient lighting and the stage lighting. Since we do not know what that is like, there is really no way to tell you what you will need exactly. Most likely you will be shooting available light if shooting during the performance. Hopefully the auditorium is very dark and the stage is very bright.
You will want both close ups and wide angle shots and a very fast lens and high shutter speed, possibly a high ISO if the children are moving and depending on the stage lighting. Stage lighting changes so you need to adapt quickly. If shooting during rehearsal, you can use a flash possibly. This can stop the action even at lower shutter speeds if the lighting is right. However, using a flash may wash out the ambient lighting which may be essential to the scene. Using the flash manually will help with that. Make sure you go to the rehearsals so you know what is happening then ask to shoot during the full dress/lighting rehearsals. Review your shots and shoot another full rehearsal. Hopefully the stage is bright enough that you do not need a flash. If you get enough good shots during the rehearsals then the night of the performance you can relax and watch the show.

Stay within your budget, compare reviews, make a list of what you expect the camera to do, make a list of the type of photos you want to take, determine where you want to head with your photography ( casual shooter, hobbyist, maybe a career) go to a camera store and play and hold them in your hands, some will feel comfy , some will not. Then pick the one you will use most and feel comfortable with.

You did not state your background in photography. The couple must trust you or they would not have asked you. I would like to know more about your background but since you are asking what is needed, Im going to assume you do not have a lot of experience shooting events or people? If you state your background and type of equipment you have, you may get a better answer to your question.

I shoot within the first week, after that you can expect the baby to be awake most of the time. (You can still shoot in the 1st 2 weeks but the baby will be more alert and wanting to stretch more). Have an assistant to help pose the baby and never take your hands off the baby without someone being right there. Feed right before, wait for a diaper change and then go for it. Work fast, keep the room warm and make sure mom is near by to nurse if needed. Supply mom with plenty of fluids and keep her comfy as well.

I love my Sony 7II. Its light weight and the 5 way stabilization is awesome. The batteries are not really an issue. I carry 3 with me for long shoots, they are cheap enough and charge up fast. I have to admit it bugged me at first but now I dont even think about it. I was a Nikon shooter for 40 years, last year I sold everything and went mirroless.

Yes, if you already have it or are getting a good deal on one. Otherwise, if you want to stay with Nikon, the 7100, D600, D610 or any Nikon that will fit your budget will work. The newer the model the better features sometime. Sony "a series" are a good choice as well, the a6000, a6500 are small, light weight and very affordable. Much newer technology than the Nikon 5500 for about the same price.

The perfect camera is the one you enjoy using, know everything about and can't live without it.

how big do you need it to be? You can use white foam core (24 x 36 or 36 x 42 inches) or go to you local building supply store, buy insulation board and paint it white, if you need something larger. Buy 2 and tape them together so that the form a V and they will then stand alone. You can also use these to reflect light as well.

Been there done that! Sold all my Nikon gear that I had for 40 some years and now own 3 Sonys.

2 speed lights and some type of diffuser will do. For diffusers, you can do a few different things. A shoot through umbrella is about 20.00 and works really well, a soft box of some kind, if you have a 5 in 1 reflector, they usually have a diffuser and you can shoot through that, or check out Gary Fongs diffuser, he uses it for portraits all the time. The speed light for used for the hair light does not need to be diffused only the main light.

Think of it this way, you gave them your negatives without developing them into prints. When you shoot RAW, you need to do some post processing just as you would in a darkroom with film.

being a novice look at the Sony 6000 series. 6000, 6300 and 6500 which ever fits your budget. What is great about this is in the view finder and the LCD screen, you will see exactly what you will get before shooting the photo. They are light weight and travel well. Babies move fast and are unpredictable, you only have one chance to get it right so seeing it before you push the button is an awesome feature. They also shoot great video.

It depends on what you want to do with the photos once you get back.

If you are in a public place, you have the right to photograph anyone including children. However, common sense has to come into play and overrules the right to photograph. I do carry a release but have never used it. If its an adult, I shoot, if its anyone under the age of 18, I ask them or their parents or both. If I ask, I also ask for their email address and send them the photo sized for internet sharing. If I intend to sell the photo, I give them a full sized file or print of the photo.

I only use it when the natural light is constantly changing from cloudy to sunny and back again.