Daria,
I can't remember about the pagers.... Yes, Princess has kid-friendly production shows. They also have ratings (G, PG, PG-13, etc) to help put the content of the other shows in perspective (the comedians, and other evening shows).
Princess divides the kids into age appropriate groups, as does Celebrity. There is a "Navigator" on both ships for the different age groups.
IMHO having sailed Celebrity twice, yes, Celebrity is a bit snooty. Things like, on formal night you HAVE to stay in your formal dress everywhere on ship (even if you choose to eat in the informal dining, you can't be out and about attending the shows in the informal dress), or you could be turned away from the show, etc. There are fashion police out there on Celebrity ships. It makes it a bit more difficult when traveling with kids. If you would like to get a flavor of the Celebrity cruise clientele, look at the cruisecritic.com message boards.
The cabins for 4 on Celebrity are EXTREMELY small. Much smaller than Princess. We had a balcony cabin on the Constellation, which was only 170 sq. ft. The larger balcony cabins (190 sq. ft.) won't accomodate 4. The Princess cabins are much nicer.
Also, re: food. Princess has a pasta course with each dinner offering which my kids truly enjoyed. They ordered it many times for their main course. Celebrity does not have that option. Our food service on our Mercury cruise was terrible (never even learned the waiter's name - we didn't see him much), as was the food. The food on the 14-day Constellation cruise was good (about equal to Princess, without the pasta course) and the service was great. Luck of the draw on what waitstaff you get.
My kids also enjoyed the 24 hour food service on Princess. You could pick up a few items at the buffet at any time for a snack. This doesn't exist on Celebrity, unless you use room service. When the kids ordered a snack from room service in the evening, it took from 1 hour to 1 1/2 hours for it to be delivered (sometimes being delivered after they went to bed).
Good luck with your decision.